An elegant way to define a sequence
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I am trying to define a sequence.
The first few terms of the sequence are:
$2,5,13,43,61$
Not yet found other terms because I am working with paper and pen, no software.
Why the first term is $5$?
Let be $pi(x)$ the celebrated prime counting function.
Well 5-$pi(5)$=$5-3$=2 which is a prime.
If we repeat the same thing with the new prime $2$, we have 2-$pi(2)=1$, which is not a prime. So starting the sequence from prime $5$, we have the cycle $5rightarrow 2rightarrow 1$. The arrows stop when a not prime is reached. No prime below $5$ has a longer cycle. Infact starting for example from $3$ you get $3-pi(3)=1$, which is not prime so the cycle is simply $3rightarrow 1$.
The second term of the sequence is $13$ because below $13$ no other prime has a larger cycle. Infact $13-pi(13)=7$, which is prime. Then $7-pi(7)=3$, which is prime and eventually $3-pi(3)=1$, which is not prime. So the cycle is $13rightarrow 7rightarrow 3rightarrow 1$
The cycle for 43 is longer so it is the third term of the above sequence.
Could you suggest to me a nice and elegant definition for this sequence: $5,13,43,61...$ (I don't know if it is infinite)
Could you find other terms with Pari if you want?
number-theory
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|
show 8 more comments
$begingroup$
I am trying to define a sequence.
The first few terms of the sequence are:
$2,5,13,43,61$
Not yet found other terms because I am working with paper and pen, no software.
Why the first term is $5$?
Let be $pi(x)$ the celebrated prime counting function.
Well 5-$pi(5)$=$5-3$=2 which is a prime.
If we repeat the same thing with the new prime $2$, we have 2-$pi(2)=1$, which is not a prime. So starting the sequence from prime $5$, we have the cycle $5rightarrow 2rightarrow 1$. The arrows stop when a not prime is reached. No prime below $5$ has a longer cycle. Infact starting for example from $3$ you get $3-pi(3)=1$, which is not prime so the cycle is simply $3rightarrow 1$.
The second term of the sequence is $13$ because below $13$ no other prime has a larger cycle. Infact $13-pi(13)=7$, which is prime. Then $7-pi(7)=3$, which is prime and eventually $3-pi(3)=1$, which is not prime. So the cycle is $13rightarrow 7rightarrow 3rightarrow 1$
The cycle for 43 is longer so it is the third term of the above sequence.
Could you suggest to me a nice and elegant definition for this sequence: $5,13,43,61...$ (I don't know if it is infinite)
Could you find other terms with Pari if you want?
number-theory
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$begingroup$
@Barry Cipra any idea?
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– homunculus
Apr 1 at 14:12
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oeis.org/A147259
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– Don Thousand
Apr 1 at 14:12
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@Don Thousand are you sure is that?
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– homunculus
Apr 1 at 14:15
1
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The next terms in the sequence are $14897$ and $377942237 $.
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– Chip Hurst
Apr 1 at 18:59
2
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Chip Hurst record goes like this: $$377942237 mapsto 357721207 mapsto 338525531 mapsto 320305991 mapsto 303015169 mapsto 286608383 mapsto 271043027 mapsto 256278002$$
$endgroup$
– Jeppe Stig Nielsen
Apr 1 at 21:31
|
show 8 more comments
$begingroup$
I am trying to define a sequence.
The first few terms of the sequence are:
$2,5,13,43,61$
Not yet found other terms because I am working with paper and pen, no software.
Why the first term is $5$?
Let be $pi(x)$ the celebrated prime counting function.
Well 5-$pi(5)$=$5-3$=2 which is a prime.
If we repeat the same thing with the new prime $2$, we have 2-$pi(2)=1$, which is not a prime. So starting the sequence from prime $5$, we have the cycle $5rightarrow 2rightarrow 1$. The arrows stop when a not prime is reached. No prime below $5$ has a longer cycle. Infact starting for example from $3$ you get $3-pi(3)=1$, which is not prime so the cycle is simply $3rightarrow 1$.
The second term of the sequence is $13$ because below $13$ no other prime has a larger cycle. Infact $13-pi(13)=7$, which is prime. Then $7-pi(7)=3$, which is prime and eventually $3-pi(3)=1$, which is not prime. So the cycle is $13rightarrow 7rightarrow 3rightarrow 1$
The cycle for 43 is longer so it is the third term of the above sequence.
Could you suggest to me a nice and elegant definition for this sequence: $5,13,43,61...$ (I don't know if it is infinite)
Could you find other terms with Pari if you want?
number-theory
$endgroup$
I am trying to define a sequence.
The first few terms of the sequence are:
$2,5,13,43,61$
Not yet found other terms because I am working with paper and pen, no software.
Why the first term is $5$?
Let be $pi(x)$ the celebrated prime counting function.
Well 5-$pi(5)$=$5-3$=2 which is a prime.
If we repeat the same thing with the new prime $2$, we have 2-$pi(2)=1$, which is not a prime. So starting the sequence from prime $5$, we have the cycle $5rightarrow 2rightarrow 1$. The arrows stop when a not prime is reached. No prime below $5$ has a longer cycle. Infact starting for example from $3$ you get $3-pi(3)=1$, which is not prime so the cycle is simply $3rightarrow 1$.
The second term of the sequence is $13$ because below $13$ no other prime has a larger cycle. Infact $13-pi(13)=7$, which is prime. Then $7-pi(7)=3$, which is prime and eventually $3-pi(3)=1$, which is not prime. So the cycle is $13rightarrow 7rightarrow 3rightarrow 1$
The cycle for 43 is longer so it is the third term of the above sequence.
Could you suggest to me a nice and elegant definition for this sequence: $5,13,43,61...$ (I don't know if it is infinite)
Could you find other terms with Pari if you want?
number-theory
number-theory
edited Apr 1 at 20:29
homunculus
asked Apr 1 at 13:52
homunculushomunculus
1919
1919
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@Barry Cipra any idea?
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– homunculus
Apr 1 at 14:12
$begingroup$
oeis.org/A147259
$endgroup$
– Don Thousand
Apr 1 at 14:12
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@Don Thousand are you sure is that?
$endgroup$
– homunculus
Apr 1 at 14:15
1
$begingroup$
The next terms in the sequence are $14897$ and $377942237 $.
$endgroup$
– Chip Hurst
Apr 1 at 18:59
2
$begingroup$
Chip Hurst record goes like this: $$377942237 mapsto 357721207 mapsto 338525531 mapsto 320305991 mapsto 303015169 mapsto 286608383 mapsto 271043027 mapsto 256278002$$
$endgroup$
– Jeppe Stig Nielsen
Apr 1 at 21:31
|
show 8 more comments
$begingroup$
@Barry Cipra any idea?
$endgroup$
– homunculus
Apr 1 at 14:12
$begingroup$
oeis.org/A147259
$endgroup$
– Don Thousand
Apr 1 at 14:12
$begingroup$
@Don Thousand are you sure is that?
$endgroup$
– homunculus
Apr 1 at 14:15
1
$begingroup$
The next terms in the sequence are $14897$ and $377942237 $.
$endgroup$
– Chip Hurst
Apr 1 at 18:59
2
$begingroup$
Chip Hurst record goes like this: $$377942237 mapsto 357721207 mapsto 338525531 mapsto 320305991 mapsto 303015169 mapsto 286608383 mapsto 271043027 mapsto 256278002$$
$endgroup$
– Jeppe Stig Nielsen
Apr 1 at 21:31
$begingroup$
@Barry Cipra any idea?
$endgroup$
– homunculus
Apr 1 at 14:12
$begingroup$
@Barry Cipra any idea?
$endgroup$
– homunculus
Apr 1 at 14:12
$begingroup$
oeis.org/A147259
$endgroup$
– Don Thousand
Apr 1 at 14:12
$begingroup$
oeis.org/A147259
$endgroup$
– Don Thousand
Apr 1 at 14:12
$begingroup$
@Don Thousand are you sure is that?
$endgroup$
– homunculus
Apr 1 at 14:15
$begingroup$
@Don Thousand are you sure is that?
$endgroup$
– homunculus
Apr 1 at 14:15
1
1
$begingroup$
The next terms in the sequence are $14897$ and $377942237 $.
$endgroup$
– Chip Hurst
Apr 1 at 18:59
$begingroup$
The next terms in the sequence are $14897$ and $377942237 $.
$endgroup$
– Chip Hurst
Apr 1 at 18:59
2
2
$begingroup$
Chip Hurst record goes like this: $$377942237 mapsto 357721207 mapsto 338525531 mapsto 320305991 mapsto 303015169 mapsto 286608383 mapsto 271043027 mapsto 256278002$$
$endgroup$
– Jeppe Stig Nielsen
Apr 1 at 21:31
$begingroup$
Chip Hurst record goes like this: $$377942237 mapsto 357721207 mapsto 338525531 mapsto 320305991 mapsto 303015169 mapsto 286608383 mapsto 271043027 mapsto 256278002$$
$endgroup$
– Jeppe Stig Nielsen
Apr 1 at 21:31
|
show 8 more comments
3 Answers
3
active
oldest
votes
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I believe your sequence continues forever but grows quickly. If $n$ is large, the density of primes around $n$ is $log n$. Since $log n$ is so much smaller than $n$, the chance a random $n$ has $k$ arrows is about $frac 1{(log n)^{k+1}}$. The expected number of sequences of length $k$ above $10^{12},$ say, is then $int_{10^{12}}^infty frac {dn}{(log n)^{k+1}}$. This diverges because $(log n)^k$ becomes less than $n$ for $n$ large enough and we know the integral of $frac 1n$ diverges. Each subtraction is only of order $frac n{log n}$, which is small compared to $n$ and the log will not change much.
If we ask what length of sequence we expect to find among the $12$ digit numbers, we note that the log of these numbers is about $29$ and that $29^{8.5} approx 3cdot 10^{12}$. We would expect to find some sequences of $7$ arrows, maybe $8$ or $9$, and be surprised at $10$ or more. For $100$ digit numbers, the log is about $231$ and $231^{42.5} approx 3cdot 10^{100}$, so we would expect some sequences of length $40$ or $41$ among the $100$ digit numbers.
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thanks but a way to define the sequence? I am still thinkig about an elegant way to define it?
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– homunculus
Apr 1 at 14:53
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You have defined it nicely. Given $n$, see how many steps of primes you get and call it $f(n)$. Your sequence are then new maxima of $f(n)$. I strongly doubt there is a way other than searching to find the sixth or tenth term.
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– Ross Millikan
Apr 1 at 15:05
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only even indexed primes after the first entry.
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– Roddy MacPhee
Apr 1 at 16:24
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@Ross Millikan but not all primes p ends the sequence with 1, isn't?
$endgroup$
– homunculus
Apr 1 at 21:28
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Formally, you can define your sequence like this: Let $phi$ be given by $$phi(p)=begin{cases}p-pi(p), & text{if $p$ is prime}\ text{undefined}, & text{otherwise}end{cases}$$ For each prime $p$, let $f(p)$ denote the maximal number of times you can iterate $phi$ starting from $p$. For example $f(43)=4$ because we get the four-arrow chain $$43mapsto 29mapsto 19mapsto 11mapsto 6$$ and you cannot go on because $6$ is not prime. Then the sequence is defined as the $p$ for which $f(p)$ is record high. So a $p$ is in the sequence iff $f(p)$ is strictly greater than $f(q)$ for all $q<p$.
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– Jeppe Stig Nielsen
Apr 1 at 22:21
add a comment |
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`my(a=0,b=0);forprime(x=1,50000,y=x;while(isprime(y-primepi(y)),y-=primepi(y);b++);if(b>a,a=b;print(x));b=0)`
produces 14897 as the next one. Then no more below 500000. There's not too much to say except primes in the sequence will be primes at even indices after the first, simply because most primes are more than 2 away from their indices.
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This supports my claim that they will grow rapidly. Thanks
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– Ross Millikan
Apr 1 at 16:37
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can we get it to support valuation of 2 for the index ? so far they are all valuation 1. If that continues checking only every 4th prime is possible.
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– Roddy MacPhee
Apr 1 at 17:03
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I note that your code has 50,000, but the text below has 500,000. Did you check it to 50,000 or 500,000?
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– Paul Sinclair
Apr 1 at 17:06
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The later (in fact tried as high as 700,000) but only after posting the code.
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– Roddy MacPhee
Apr 1 at 17:09
add a comment |
$begingroup$
Using $S$ to denote the sequence you're trying to define, one may do so in
terms of two auxiliary functions $N$ and $L$, where $N$ assigns to every prime number $x$ a sequence whose first term, denoted by $(N(x))(0)$$^*$, is $x$ itself, and each next term, denoted by $(N(x))(n + 1)$, is given by $(N(x))(n) - pi((N(x))(n))$, and $L$ is the function which gives the number of terms of a sequence returned by $N$ up to when the first non-prime term is reached. $S$ is then defined to be such that the first term equals $5$, and given any term $S(n)$, the next term in the sequence is then the smallest prime number $p$ such that $L(N(p)) > L(N(S(n)))$.
In formula:
$S(0) = 5$
$S(n + 1) = langledownarrow p : p in mathbb{P} : L(N(p)) > L(N(S(n)))rangle$
$(N(x))(0) = x$
$(N(x))(n + 1) = (N(x))(n) - pi((N(x))(n))$
$L(N(x)) = langledownarrow n : n in mathbb{N} : (N(x))(n) notin
mathbb{P}rangle$
The notation $langledownarrow x : R(x) : T(x)rangle$ here denotes the minimum element $x$ that satisfies $T(x)$ from the set of all elements satisfying $R(x)$. $R(x)$ and $T(x)$ denote arbitrary predicates (i. e. boolean-valued functions) which generally depend on $x$.
$^*$Note: Here we use the definition that a sequence is any function whose domain consists of either all natural numbers or all natural numbers $n$ such that $0 le n lt m$ for arbitrary natural constant $m$. We admit $0$ as the smallest natural number.
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I have edited this answer many times already, but I think I'm done now. I have performed a major clean-up on it by leaving the precise delineation of the domains and codomains of $S$, $N(x)$ and $L$ unspecified, as that degree of overspecification is completely irrelevant to the problem.
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– lucasb
yesterday
add a comment |
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3 Answers
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3 Answers
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oldest
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$begingroup$
I believe your sequence continues forever but grows quickly. If $n$ is large, the density of primes around $n$ is $log n$. Since $log n$ is so much smaller than $n$, the chance a random $n$ has $k$ arrows is about $frac 1{(log n)^{k+1}}$. The expected number of sequences of length $k$ above $10^{12},$ say, is then $int_{10^{12}}^infty frac {dn}{(log n)^{k+1}}$. This diverges because $(log n)^k$ becomes less than $n$ for $n$ large enough and we know the integral of $frac 1n$ diverges. Each subtraction is only of order $frac n{log n}$, which is small compared to $n$ and the log will not change much.
If we ask what length of sequence we expect to find among the $12$ digit numbers, we note that the log of these numbers is about $29$ and that $29^{8.5} approx 3cdot 10^{12}$. We would expect to find some sequences of $7$ arrows, maybe $8$ or $9$, and be surprised at $10$ or more. For $100$ digit numbers, the log is about $231$ and $231^{42.5} approx 3cdot 10^{100}$, so we would expect some sequences of length $40$ or $41$ among the $100$ digit numbers.
$endgroup$
$begingroup$
thanks but a way to define the sequence? I am still thinkig about an elegant way to define it?
$endgroup$
– homunculus
Apr 1 at 14:53
$begingroup$
You have defined it nicely. Given $n$, see how many steps of primes you get and call it $f(n)$. Your sequence are then new maxima of $f(n)$. I strongly doubt there is a way other than searching to find the sixth or tenth term.
$endgroup$
– Ross Millikan
Apr 1 at 15:05
$begingroup$
only even indexed primes after the first entry.
$endgroup$
– Roddy MacPhee
Apr 1 at 16:24
$begingroup$
@Ross Millikan but not all primes p ends the sequence with 1, isn't?
$endgroup$
– homunculus
Apr 1 at 21:28
$begingroup$
Formally, you can define your sequence like this: Let $phi$ be given by $$phi(p)=begin{cases}p-pi(p), & text{if $p$ is prime}\ text{undefined}, & text{otherwise}end{cases}$$ For each prime $p$, let $f(p)$ denote the maximal number of times you can iterate $phi$ starting from $p$. For example $f(43)=4$ because we get the four-arrow chain $$43mapsto 29mapsto 19mapsto 11mapsto 6$$ and you cannot go on because $6$ is not prime. Then the sequence is defined as the $p$ for which $f(p)$ is record high. So a $p$ is in the sequence iff $f(p)$ is strictly greater than $f(q)$ for all $q<p$.
$endgroup$
– Jeppe Stig Nielsen
Apr 1 at 22:21
add a comment |
$begingroup$
I believe your sequence continues forever but grows quickly. If $n$ is large, the density of primes around $n$ is $log n$. Since $log n$ is so much smaller than $n$, the chance a random $n$ has $k$ arrows is about $frac 1{(log n)^{k+1}}$. The expected number of sequences of length $k$ above $10^{12},$ say, is then $int_{10^{12}}^infty frac {dn}{(log n)^{k+1}}$. This diverges because $(log n)^k$ becomes less than $n$ for $n$ large enough and we know the integral of $frac 1n$ diverges. Each subtraction is only of order $frac n{log n}$, which is small compared to $n$ and the log will not change much.
If we ask what length of sequence we expect to find among the $12$ digit numbers, we note that the log of these numbers is about $29$ and that $29^{8.5} approx 3cdot 10^{12}$. We would expect to find some sequences of $7$ arrows, maybe $8$ or $9$, and be surprised at $10$ or more. For $100$ digit numbers, the log is about $231$ and $231^{42.5} approx 3cdot 10^{100}$, so we would expect some sequences of length $40$ or $41$ among the $100$ digit numbers.
$endgroup$
$begingroup$
thanks but a way to define the sequence? I am still thinkig about an elegant way to define it?
$endgroup$
– homunculus
Apr 1 at 14:53
$begingroup$
You have defined it nicely. Given $n$, see how many steps of primes you get and call it $f(n)$. Your sequence are then new maxima of $f(n)$. I strongly doubt there is a way other than searching to find the sixth or tenth term.
$endgroup$
– Ross Millikan
Apr 1 at 15:05
$begingroup$
only even indexed primes after the first entry.
$endgroup$
– Roddy MacPhee
Apr 1 at 16:24
$begingroup$
@Ross Millikan but not all primes p ends the sequence with 1, isn't?
$endgroup$
– homunculus
Apr 1 at 21:28
$begingroup$
Formally, you can define your sequence like this: Let $phi$ be given by $$phi(p)=begin{cases}p-pi(p), & text{if $p$ is prime}\ text{undefined}, & text{otherwise}end{cases}$$ For each prime $p$, let $f(p)$ denote the maximal number of times you can iterate $phi$ starting from $p$. For example $f(43)=4$ because we get the four-arrow chain $$43mapsto 29mapsto 19mapsto 11mapsto 6$$ and you cannot go on because $6$ is not prime. Then the sequence is defined as the $p$ for which $f(p)$ is record high. So a $p$ is in the sequence iff $f(p)$ is strictly greater than $f(q)$ for all $q<p$.
$endgroup$
– Jeppe Stig Nielsen
Apr 1 at 22:21
add a comment |
$begingroup$
I believe your sequence continues forever but grows quickly. If $n$ is large, the density of primes around $n$ is $log n$. Since $log n$ is so much smaller than $n$, the chance a random $n$ has $k$ arrows is about $frac 1{(log n)^{k+1}}$. The expected number of sequences of length $k$ above $10^{12},$ say, is then $int_{10^{12}}^infty frac {dn}{(log n)^{k+1}}$. This diverges because $(log n)^k$ becomes less than $n$ for $n$ large enough and we know the integral of $frac 1n$ diverges. Each subtraction is only of order $frac n{log n}$, which is small compared to $n$ and the log will not change much.
If we ask what length of sequence we expect to find among the $12$ digit numbers, we note that the log of these numbers is about $29$ and that $29^{8.5} approx 3cdot 10^{12}$. We would expect to find some sequences of $7$ arrows, maybe $8$ or $9$, and be surprised at $10$ or more. For $100$ digit numbers, the log is about $231$ and $231^{42.5} approx 3cdot 10^{100}$, so we would expect some sequences of length $40$ or $41$ among the $100$ digit numbers.
$endgroup$
I believe your sequence continues forever but grows quickly. If $n$ is large, the density of primes around $n$ is $log n$. Since $log n$ is so much smaller than $n$, the chance a random $n$ has $k$ arrows is about $frac 1{(log n)^{k+1}}$. The expected number of sequences of length $k$ above $10^{12},$ say, is then $int_{10^{12}}^infty frac {dn}{(log n)^{k+1}}$. This diverges because $(log n)^k$ becomes less than $n$ for $n$ large enough and we know the integral of $frac 1n$ diverges. Each subtraction is only of order $frac n{log n}$, which is small compared to $n$ and the log will not change much.
If we ask what length of sequence we expect to find among the $12$ digit numbers, we note that the log of these numbers is about $29$ and that $29^{8.5} approx 3cdot 10^{12}$. We would expect to find some sequences of $7$ arrows, maybe $8$ or $9$, and be surprised at $10$ or more. For $100$ digit numbers, the log is about $231$ and $231^{42.5} approx 3cdot 10^{100}$, so we would expect some sequences of length $40$ or $41$ among the $100$ digit numbers.
answered Apr 1 at 14:43
Ross MillikanRoss Millikan
301k24200375
301k24200375
$begingroup$
thanks but a way to define the sequence? I am still thinkig about an elegant way to define it?
$endgroup$
– homunculus
Apr 1 at 14:53
$begingroup$
You have defined it nicely. Given $n$, see how many steps of primes you get and call it $f(n)$. Your sequence are then new maxima of $f(n)$. I strongly doubt there is a way other than searching to find the sixth or tenth term.
$endgroup$
– Ross Millikan
Apr 1 at 15:05
$begingroup$
only even indexed primes after the first entry.
$endgroup$
– Roddy MacPhee
Apr 1 at 16:24
$begingroup$
@Ross Millikan but not all primes p ends the sequence with 1, isn't?
$endgroup$
– homunculus
Apr 1 at 21:28
$begingroup$
Formally, you can define your sequence like this: Let $phi$ be given by $$phi(p)=begin{cases}p-pi(p), & text{if $p$ is prime}\ text{undefined}, & text{otherwise}end{cases}$$ For each prime $p$, let $f(p)$ denote the maximal number of times you can iterate $phi$ starting from $p$. For example $f(43)=4$ because we get the four-arrow chain $$43mapsto 29mapsto 19mapsto 11mapsto 6$$ and you cannot go on because $6$ is not prime. Then the sequence is defined as the $p$ for which $f(p)$ is record high. So a $p$ is in the sequence iff $f(p)$ is strictly greater than $f(q)$ for all $q<p$.
$endgroup$
– Jeppe Stig Nielsen
Apr 1 at 22:21
add a comment |
$begingroup$
thanks but a way to define the sequence? I am still thinkig about an elegant way to define it?
$endgroup$
– homunculus
Apr 1 at 14:53
$begingroup$
You have defined it nicely. Given $n$, see how many steps of primes you get and call it $f(n)$. Your sequence are then new maxima of $f(n)$. I strongly doubt there is a way other than searching to find the sixth or tenth term.
$endgroup$
– Ross Millikan
Apr 1 at 15:05
$begingroup$
only even indexed primes after the first entry.
$endgroup$
– Roddy MacPhee
Apr 1 at 16:24
$begingroup$
@Ross Millikan but not all primes p ends the sequence with 1, isn't?
$endgroup$
– homunculus
Apr 1 at 21:28
$begingroup$
Formally, you can define your sequence like this: Let $phi$ be given by $$phi(p)=begin{cases}p-pi(p), & text{if $p$ is prime}\ text{undefined}, & text{otherwise}end{cases}$$ For each prime $p$, let $f(p)$ denote the maximal number of times you can iterate $phi$ starting from $p$. For example $f(43)=4$ because we get the four-arrow chain $$43mapsto 29mapsto 19mapsto 11mapsto 6$$ and you cannot go on because $6$ is not prime. Then the sequence is defined as the $p$ for which $f(p)$ is record high. So a $p$ is in the sequence iff $f(p)$ is strictly greater than $f(q)$ for all $q<p$.
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– Jeppe Stig Nielsen
Apr 1 at 22:21
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thanks but a way to define the sequence? I am still thinkig about an elegant way to define it?
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– homunculus
Apr 1 at 14:53
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thanks but a way to define the sequence? I am still thinkig about an elegant way to define it?
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– homunculus
Apr 1 at 14:53
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You have defined it nicely. Given $n$, see how many steps of primes you get and call it $f(n)$. Your sequence are then new maxima of $f(n)$. I strongly doubt there is a way other than searching to find the sixth or tenth term.
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– Ross Millikan
Apr 1 at 15:05
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You have defined it nicely. Given $n$, see how many steps of primes you get and call it $f(n)$. Your sequence are then new maxima of $f(n)$. I strongly doubt there is a way other than searching to find the sixth or tenth term.
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– Ross Millikan
Apr 1 at 15:05
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only even indexed primes after the first entry.
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– Roddy MacPhee
Apr 1 at 16:24
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only even indexed primes after the first entry.
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– Roddy MacPhee
Apr 1 at 16:24
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@Ross Millikan but not all primes p ends the sequence with 1, isn't?
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– homunculus
Apr 1 at 21:28
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@Ross Millikan but not all primes p ends the sequence with 1, isn't?
$endgroup$
– homunculus
Apr 1 at 21:28
$begingroup$
Formally, you can define your sequence like this: Let $phi$ be given by $$phi(p)=begin{cases}p-pi(p), & text{if $p$ is prime}\ text{undefined}, & text{otherwise}end{cases}$$ For each prime $p$, let $f(p)$ denote the maximal number of times you can iterate $phi$ starting from $p$. For example $f(43)=4$ because we get the four-arrow chain $$43mapsto 29mapsto 19mapsto 11mapsto 6$$ and you cannot go on because $6$ is not prime. Then the sequence is defined as the $p$ for which $f(p)$ is record high. So a $p$ is in the sequence iff $f(p)$ is strictly greater than $f(q)$ for all $q<p$.
$endgroup$
– Jeppe Stig Nielsen
Apr 1 at 22:21
$begingroup$
Formally, you can define your sequence like this: Let $phi$ be given by $$phi(p)=begin{cases}p-pi(p), & text{if $p$ is prime}\ text{undefined}, & text{otherwise}end{cases}$$ For each prime $p$, let $f(p)$ denote the maximal number of times you can iterate $phi$ starting from $p$. For example $f(43)=4$ because we get the four-arrow chain $$43mapsto 29mapsto 19mapsto 11mapsto 6$$ and you cannot go on because $6$ is not prime. Then the sequence is defined as the $p$ for which $f(p)$ is record high. So a $p$ is in the sequence iff $f(p)$ is strictly greater than $f(q)$ for all $q<p$.
$endgroup$
– Jeppe Stig Nielsen
Apr 1 at 22:21
add a comment |
$begingroup$
`my(a=0,b=0);forprime(x=1,50000,y=x;while(isprime(y-primepi(y)),y-=primepi(y);b++);if(b>a,a=b;print(x));b=0)`
produces 14897 as the next one. Then no more below 500000. There's not too much to say except primes in the sequence will be primes at even indices after the first, simply because most primes are more than 2 away from their indices.
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This supports my claim that they will grow rapidly. Thanks
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– Ross Millikan
Apr 1 at 16:37
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can we get it to support valuation of 2 for the index ? so far they are all valuation 1. If that continues checking only every 4th prime is possible.
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– Roddy MacPhee
Apr 1 at 17:03
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I note that your code has 50,000, but the text below has 500,000. Did you check it to 50,000 or 500,000?
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– Paul Sinclair
Apr 1 at 17:06
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The later (in fact tried as high as 700,000) but only after posting the code.
$endgroup$
– Roddy MacPhee
Apr 1 at 17:09
add a comment |
$begingroup$
`my(a=0,b=0);forprime(x=1,50000,y=x;while(isprime(y-primepi(y)),y-=primepi(y);b++);if(b>a,a=b;print(x));b=0)`
produces 14897 as the next one. Then no more below 500000. There's not too much to say except primes in the sequence will be primes at even indices after the first, simply because most primes are more than 2 away from their indices.
$endgroup$
$begingroup$
This supports my claim that they will grow rapidly. Thanks
$endgroup$
– Ross Millikan
Apr 1 at 16:37
$begingroup$
can we get it to support valuation of 2 for the index ? so far they are all valuation 1. If that continues checking only every 4th prime is possible.
$endgroup$
– Roddy MacPhee
Apr 1 at 17:03
$begingroup$
I note that your code has 50,000, but the text below has 500,000. Did you check it to 50,000 or 500,000?
$endgroup$
– Paul Sinclair
Apr 1 at 17:06
$begingroup$
The later (in fact tried as high as 700,000) but only after posting the code.
$endgroup$
– Roddy MacPhee
Apr 1 at 17:09
add a comment |
$begingroup$
`my(a=0,b=0);forprime(x=1,50000,y=x;while(isprime(y-primepi(y)),y-=primepi(y);b++);if(b>a,a=b;print(x));b=0)`
produces 14897 as the next one. Then no more below 500000. There's not too much to say except primes in the sequence will be primes at even indices after the first, simply because most primes are more than 2 away from their indices.
$endgroup$
`my(a=0,b=0);forprime(x=1,50000,y=x;while(isprime(y-primepi(y)),y-=primepi(y);b++);if(b>a,a=b;print(x));b=0)`
produces 14897 as the next one. Then no more below 500000. There's not too much to say except primes in the sequence will be primes at even indices after the first, simply because most primes are more than 2 away from their indices.
answered Apr 1 at 15:52
Roddy MacPheeRoddy MacPhee
648118
648118
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This supports my claim that they will grow rapidly. Thanks
$endgroup$
– Ross Millikan
Apr 1 at 16:37
$begingroup$
can we get it to support valuation of 2 for the index ? so far they are all valuation 1. If that continues checking only every 4th prime is possible.
$endgroup$
– Roddy MacPhee
Apr 1 at 17:03
$begingroup$
I note that your code has 50,000, but the text below has 500,000. Did you check it to 50,000 or 500,000?
$endgroup$
– Paul Sinclair
Apr 1 at 17:06
$begingroup$
The later (in fact tried as high as 700,000) but only after posting the code.
$endgroup$
– Roddy MacPhee
Apr 1 at 17:09
add a comment |
$begingroup$
This supports my claim that they will grow rapidly. Thanks
$endgroup$
– Ross Millikan
Apr 1 at 16:37
$begingroup$
can we get it to support valuation of 2 for the index ? so far they are all valuation 1. If that continues checking only every 4th prime is possible.
$endgroup$
– Roddy MacPhee
Apr 1 at 17:03
$begingroup$
I note that your code has 50,000, but the text below has 500,000. Did you check it to 50,000 or 500,000?
$endgroup$
– Paul Sinclair
Apr 1 at 17:06
$begingroup$
The later (in fact tried as high as 700,000) but only after posting the code.
$endgroup$
– Roddy MacPhee
Apr 1 at 17:09
$begingroup$
This supports my claim that they will grow rapidly. Thanks
$endgroup$
– Ross Millikan
Apr 1 at 16:37
$begingroup$
This supports my claim that they will grow rapidly. Thanks
$endgroup$
– Ross Millikan
Apr 1 at 16:37
$begingroup$
can we get it to support valuation of 2 for the index ? so far they are all valuation 1. If that continues checking only every 4th prime is possible.
$endgroup$
– Roddy MacPhee
Apr 1 at 17:03
$begingroup$
can we get it to support valuation of 2 for the index ? so far they are all valuation 1. If that continues checking only every 4th prime is possible.
$endgroup$
– Roddy MacPhee
Apr 1 at 17:03
$begingroup$
I note that your code has 50,000, but the text below has 500,000. Did you check it to 50,000 or 500,000?
$endgroup$
– Paul Sinclair
Apr 1 at 17:06
$begingroup$
I note that your code has 50,000, but the text below has 500,000. Did you check it to 50,000 or 500,000?
$endgroup$
– Paul Sinclair
Apr 1 at 17:06
$begingroup$
The later (in fact tried as high as 700,000) but only after posting the code.
$endgroup$
– Roddy MacPhee
Apr 1 at 17:09
$begingroup$
The later (in fact tried as high as 700,000) but only after posting the code.
$endgroup$
– Roddy MacPhee
Apr 1 at 17:09
add a comment |
$begingroup$
Using $S$ to denote the sequence you're trying to define, one may do so in
terms of two auxiliary functions $N$ and $L$, where $N$ assigns to every prime number $x$ a sequence whose first term, denoted by $(N(x))(0)$$^*$, is $x$ itself, and each next term, denoted by $(N(x))(n + 1)$, is given by $(N(x))(n) - pi((N(x))(n))$, and $L$ is the function which gives the number of terms of a sequence returned by $N$ up to when the first non-prime term is reached. $S$ is then defined to be such that the first term equals $5$, and given any term $S(n)$, the next term in the sequence is then the smallest prime number $p$ such that $L(N(p)) > L(N(S(n)))$.
In formula:
$S(0) = 5$
$S(n + 1) = langledownarrow p : p in mathbb{P} : L(N(p)) > L(N(S(n)))rangle$
$(N(x))(0) = x$
$(N(x))(n + 1) = (N(x))(n) - pi((N(x))(n))$
$L(N(x)) = langledownarrow n : n in mathbb{N} : (N(x))(n) notin
mathbb{P}rangle$
The notation $langledownarrow x : R(x) : T(x)rangle$ here denotes the minimum element $x$ that satisfies $T(x)$ from the set of all elements satisfying $R(x)$. $R(x)$ and $T(x)$ denote arbitrary predicates (i. e. boolean-valued functions) which generally depend on $x$.
$^*$Note: Here we use the definition that a sequence is any function whose domain consists of either all natural numbers or all natural numbers $n$ such that $0 le n lt m$ for arbitrary natural constant $m$. We admit $0$ as the smallest natural number.
$endgroup$
$begingroup$
I have edited this answer many times already, but I think I'm done now. I have performed a major clean-up on it by leaving the precise delineation of the domains and codomains of $S$, $N(x)$ and $L$ unspecified, as that degree of overspecification is completely irrelevant to the problem.
$endgroup$
– lucasb
yesterday
add a comment |
$begingroup$
Using $S$ to denote the sequence you're trying to define, one may do so in
terms of two auxiliary functions $N$ and $L$, where $N$ assigns to every prime number $x$ a sequence whose first term, denoted by $(N(x))(0)$$^*$, is $x$ itself, and each next term, denoted by $(N(x))(n + 1)$, is given by $(N(x))(n) - pi((N(x))(n))$, and $L$ is the function which gives the number of terms of a sequence returned by $N$ up to when the first non-prime term is reached. $S$ is then defined to be such that the first term equals $5$, and given any term $S(n)$, the next term in the sequence is then the smallest prime number $p$ such that $L(N(p)) > L(N(S(n)))$.
In formula:
$S(0) = 5$
$S(n + 1) = langledownarrow p : p in mathbb{P} : L(N(p)) > L(N(S(n)))rangle$
$(N(x))(0) = x$
$(N(x))(n + 1) = (N(x))(n) - pi((N(x))(n))$
$L(N(x)) = langledownarrow n : n in mathbb{N} : (N(x))(n) notin
mathbb{P}rangle$
The notation $langledownarrow x : R(x) : T(x)rangle$ here denotes the minimum element $x$ that satisfies $T(x)$ from the set of all elements satisfying $R(x)$. $R(x)$ and $T(x)$ denote arbitrary predicates (i. e. boolean-valued functions) which generally depend on $x$.
$^*$Note: Here we use the definition that a sequence is any function whose domain consists of either all natural numbers or all natural numbers $n$ such that $0 le n lt m$ for arbitrary natural constant $m$. We admit $0$ as the smallest natural number.
$endgroup$
$begingroup$
I have edited this answer many times already, but I think I'm done now. I have performed a major clean-up on it by leaving the precise delineation of the domains and codomains of $S$, $N(x)$ and $L$ unspecified, as that degree of overspecification is completely irrelevant to the problem.
$endgroup$
– lucasb
yesterday
add a comment |
$begingroup$
Using $S$ to denote the sequence you're trying to define, one may do so in
terms of two auxiliary functions $N$ and $L$, where $N$ assigns to every prime number $x$ a sequence whose first term, denoted by $(N(x))(0)$$^*$, is $x$ itself, and each next term, denoted by $(N(x))(n + 1)$, is given by $(N(x))(n) - pi((N(x))(n))$, and $L$ is the function which gives the number of terms of a sequence returned by $N$ up to when the first non-prime term is reached. $S$ is then defined to be such that the first term equals $5$, and given any term $S(n)$, the next term in the sequence is then the smallest prime number $p$ such that $L(N(p)) > L(N(S(n)))$.
In formula:
$S(0) = 5$
$S(n + 1) = langledownarrow p : p in mathbb{P} : L(N(p)) > L(N(S(n)))rangle$
$(N(x))(0) = x$
$(N(x))(n + 1) = (N(x))(n) - pi((N(x))(n))$
$L(N(x)) = langledownarrow n : n in mathbb{N} : (N(x))(n) notin
mathbb{P}rangle$
The notation $langledownarrow x : R(x) : T(x)rangle$ here denotes the minimum element $x$ that satisfies $T(x)$ from the set of all elements satisfying $R(x)$. $R(x)$ and $T(x)$ denote arbitrary predicates (i. e. boolean-valued functions) which generally depend on $x$.
$^*$Note: Here we use the definition that a sequence is any function whose domain consists of either all natural numbers or all natural numbers $n$ such that $0 le n lt m$ for arbitrary natural constant $m$. We admit $0$ as the smallest natural number.
$endgroup$
Using $S$ to denote the sequence you're trying to define, one may do so in
terms of two auxiliary functions $N$ and $L$, where $N$ assigns to every prime number $x$ a sequence whose first term, denoted by $(N(x))(0)$$^*$, is $x$ itself, and each next term, denoted by $(N(x))(n + 1)$, is given by $(N(x))(n) - pi((N(x))(n))$, and $L$ is the function which gives the number of terms of a sequence returned by $N$ up to when the first non-prime term is reached. $S$ is then defined to be such that the first term equals $5$, and given any term $S(n)$, the next term in the sequence is then the smallest prime number $p$ such that $L(N(p)) > L(N(S(n)))$.
In formula:
$S(0) = 5$
$S(n + 1) = langledownarrow p : p in mathbb{P} : L(N(p)) > L(N(S(n)))rangle$
$(N(x))(0) = x$
$(N(x))(n + 1) = (N(x))(n) - pi((N(x))(n))$
$L(N(x)) = langledownarrow n : n in mathbb{N} : (N(x))(n) notin
mathbb{P}rangle$
The notation $langledownarrow x : R(x) : T(x)rangle$ here denotes the minimum element $x$ that satisfies $T(x)$ from the set of all elements satisfying $R(x)$. $R(x)$ and $T(x)$ denote arbitrary predicates (i. e. boolean-valued functions) which generally depend on $x$.
$^*$Note: Here we use the definition that a sequence is any function whose domain consists of either all natural numbers or all natural numbers $n$ such that $0 le n lt m$ for arbitrary natural constant $m$. We admit $0$ as the smallest natural number.
edited yesterday
answered Apr 1 at 21:33
lucasblucasb
212
212
$begingroup$
I have edited this answer many times already, but I think I'm done now. I have performed a major clean-up on it by leaving the precise delineation of the domains and codomains of $S$, $N(x)$ and $L$ unspecified, as that degree of overspecification is completely irrelevant to the problem.
$endgroup$
– lucasb
yesterday
add a comment |
$begingroup$
I have edited this answer many times already, but I think I'm done now. I have performed a major clean-up on it by leaving the precise delineation of the domains and codomains of $S$, $N(x)$ and $L$ unspecified, as that degree of overspecification is completely irrelevant to the problem.
$endgroup$
– lucasb
yesterday
$begingroup$
I have edited this answer many times already, but I think I'm done now. I have performed a major clean-up on it by leaving the precise delineation of the domains and codomains of $S$, $N(x)$ and $L$ unspecified, as that degree of overspecification is completely irrelevant to the problem.
$endgroup$
– lucasb
yesterday
$begingroup$
I have edited this answer many times already, but I think I'm done now. I have performed a major clean-up on it by leaving the precise delineation of the domains and codomains of $S$, $N(x)$ and $L$ unspecified, as that degree of overspecification is completely irrelevant to the problem.
$endgroup$
– lucasb
yesterday
add a comment |
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$begingroup$
@Barry Cipra any idea?
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– homunculus
Apr 1 at 14:12
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oeis.org/A147259
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– Don Thousand
Apr 1 at 14:12
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@Don Thousand are you sure is that?
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– homunculus
Apr 1 at 14:15
1
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The next terms in the sequence are $14897$ and $377942237 $.
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– Chip Hurst
Apr 1 at 18:59
2
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Chip Hurst record goes like this: $$377942237 mapsto 357721207 mapsto 338525531 mapsto 320305991 mapsto 303015169 mapsto 286608383 mapsto 271043027 mapsto 256278002$$
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– Jeppe Stig Nielsen
Apr 1 at 21:31