A137288 -id:A137288 - cp's OEIS frontend

A181715 Length of the complete Cunningham chain of the second kind starting with prime(n).

3, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1

Offset: 1

M. F. Hasler, Nov 17 2010

nonn

Number of iterations x -> 2x-1 needed to get a composite number, when starting with prime(n).
Dickson's conjecture implies that, for every positive integer r, there exist infinitely many n such that a(n) = r. - Lorenzo Sauras Altuzarra, Feb 12 2021
a(n) is the least k such that 2^k * (prime(n)-1) + 1 is composite. Note that a(n) is well defined since 2^(p-1) * (p-1) + 1 is divisible by p for odd primes p. - Jianing Song, Nov 24 2021

			2 -> 3 -> 5 -> 9 = 3^2, so a(1) = 3 and a(2) = 2. - _Jonathan Sondow_, Oct 30 2015

T. D. Noe, Table of n, a(n) for n = 1..10000
G. Löh, Long chains of nearly doubled primes, Math. Comp., 53 (1989), 751-759.
Michael Penn, Romanian Mathematical Olympiad Problem, Youtube video, 2020.
Wikipedia, Cunningham chain

Cf. A000040, A005382, A005408, A005602, A005603, A181697, A263879, A285700, A285706, A057326, A057327, A057328, A057329, A057330, A064812.

Cf. A137288 (positions of terms > 1).

a := proc(n)
   local c, l:
   c, l := 0, ithprime(n):
   while isprime(l) do c, l := c+1, 2*l-1: od:
   c:
end: # Lorenzo Sauras Altuzarra, Feb 12 2021

Table[p = Prime[n]; cnt = 1; While[p = 2*p - 1; PrimeQ[p], cnt++]; cnt, {n, 100}] (* T. D. Noe, Jul 12 2012 *)
Table[-1 + Length@ NestWhileList[2 # - 1 &, Prime@ n, PrimeQ@ # &], {n, 98}] (* Michael De Vlieger, Apr 26 2017 *)

a(n)= n=prime(n); for(c=1,1e9, is/*pseudo*/prime(n=2*n-1) || return(c))

a(n) < prime(n) for n > 1; see Löh (1989), p. 751. - Jonathan Sondow, Oct 28 2015
max(a(n), A181697(n)) = A263879(n) for n > 2. - Jonathan Sondow, Oct 30 2015
a(n) = A285700(A000040(n)). - Antti Karttunen, Apr 26 2017

Escape clause added to definition by N. J. A. Sloane, Feb 19 2021

Escape clause deleted from definition by Jianing Song, Nov 24 2021

A278229 Least number with the prime signature of 2*prime(n) - 1.

Original entry on oeis.org

2, 2, 4, 2, 6, 4, 6, 2, 12, 6, 2, 2, 16, 6, 6, 30, 12, 4, 6, 6, 6, 2, 30, 6, 2, 6, 6, 6, 6, 36, 6, 12, 30, 2, 24, 6, 2, 12, 12, 30, 30, 4, 6, 30, 6, 2, 2, 6, 6, 2, 30, 12, 6, 6, 24, 60, 6, 2, 6, 30, 6, 60, 2, 24, 16, 6, 2, 2, 60, 6, 30, 6, 2, 6, 2, 60, 30, 6, 12, 6, 24, 4, 30, 6, 2, 30, 30, 6, 6, 12, 6, 30, 6, 12, 2, 30, 12, 6, 30, 6, 2, 30, 72, 6, 6, 2, 30, 30

Offset: 1

Antti Karttunen, Nov 19 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000040, A046523, A076274.
Cf. A137288 (positions of 2's), A005382.
Cf. also A278227, A278228, A278230.

Table[Times @@ MapIndexed[(Prime@ First@ #2)^#1 &, #] &@ If[Length@ # == 1 && #[[1, 1]] == 1, {0}, Reverse@ Sort@ #[[All, -1]]] &@ FactorInteger[ 2 Prime@ n - 1], {n, 120}] (* Michael De Vlieger, Nov 21 2016 *)

(define (A278229 n) (A046523 (+ -1 (* 2 (A000040 n)))))

a(n) = A046523(A076274(n)) = A046523((2*A000040(n))-1).

A285706 a(n) = number of iterations x -> A064216(x) needed to reach a nonprime number when starting from prime(n), a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Apr 26 2017

nonn

Length (or size for the closed cycles: [2] and [3]) of the complete "slipping Cunningham chain of the second kind" starting with prime(n). That is, at the end of every step, the next prime q = 2p-1 "slips" by one step towards smaller primes as A064989(q).

After n = 1, 2 (primes 2 & 3) differs from A181715 for the first time at n=22, where a(22) = 2, while A181715(22) = 3, prime(22) = 79.

			See examples in A285701.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000040, A064216, A064989, A181715, A246373, A285701.

Cf. A137288 (gives the positions of terms > 1 after its two initial terms).

Table[If[n <= 2, 1, -1 + Length@ NestWhileList[Apply[Times, FactorInteger[2 # - 1] /. {p_, e_} /; p > 2 :> NextPrime[p, -1]^e] &, Prime@ n, PrimeQ@ # &]], {n, 120}] (* Michael De Vlieger, Apr 26 2017 *)

A285706(n) = A285701(prime(n)); \\ The rest of code in A285701.

(define (A285706 n) (A285701 (A000040 n)))

a(n) = A285701(A000040(n)).

A023583 Greatest prime divisor of 2*prime(n)-1.

Original entry on oeis.org

3, 5, 3, 13, 7, 5, 11, 37, 5, 19, 61, 73, 3, 17, 31, 7, 13, 11, 19, 47, 29, 157, 11, 59, 193, 67, 41, 71, 31, 5, 23, 29, 13, 277, 11, 43, 313, 13, 37, 23, 17, 19, 127, 11, 131, 397, 421, 89, 151, 457, 31, 53, 37, 167, 19, 7, 179, 541, 79, 17, 113, 13, 613, 23, 5

Offset: 1

Clark Kimberling

a(n) = 2*prime(n)-1 if n is in A137288. - Robert Israel, May 19 2020

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006530, A076274, A137288.

f:= n -> max(numtheory:-factorset(2*ithprime(n)-1)):
map(f, [$1..100]); # Robert Israel, May 19 2020

a[n_] := FactorInteger[2*Prime[n]-1][[-1, 1]]; Array[a, 100] (* Amiram Eldar, Oct 27 2024 *)

a(n) = {my(f = factor(2*prime(n)-1)); f[#f~, 1];} \\ Amiram Eldar, Oct 27 2024

a(n) = A006530(A076274(n+1)). - Bernard Schott, May 20 2020

Name edited by Robert Israel, May 19 2020

A307368 a(n) is the minimal positive integer such that 2a(n)prime(n)-1 equals another prime.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 2, 1, 3, 3, 1, 1, 2, 3, 3, 2, 3, 4, 3, 2, 7, 1, 2, 8, 1, 5, 3, 3, 3, 3, 3, 2, 2, 1, 5, 6, 1, 3, 5, 2, 5, 4, 11, 4, 2, 1, 1, 4, 2, 1, 8, 3, 7, 6, 6, 2, 3, 1, 6, 2, 3, 2, 1, 5, 3, 3, 1, 1, 3, 4, 5, 3, 1, 3, 1, 2, 3, 3, 11, 4, 8, 6, 2, 4, 1, 3, 3, 3, 6, 3, 2, 5, 6, 5, 1, 2, 9, 2, 3, 4, 1, 5, 2, 3, 4, 1, 2, 2, 3

Offset: 1

Ivan N. Ianakiev, Apr 17 2019

nonn

A more general form of Rassias's conjecture states that for every positive integer a there are two primes p and q such that 2*a*p = q+1.
a(n)=1 for n in A137288. - Robert Israel, Apr 18 2019
By Dirichlet's theorem on primes in arithmetic progressions, a(n) exists. - Robert Israel, May 12 2019

Michael Th. Rassias, Problem-Solving and Selected Topics in Number Theory, Springer-Verlag, NY, 2011, pp. xi-xii.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A053989, A137288.

f:= proc(n) local k,p;
    p:= ithprime(n);
    for k from 1 do
      if isprime(2*k*p-1) then return k fi
    od
end proc:
map(f, [$1..100]); # Robert Israel, Apr 18 2019

a[n_]:=Module[{a=1},While[!PrimeQ[2*a*Prime[n]-1],a++];a];
a/@Range[110]

a(n) = my(p=prime(n)); for(k=1, oo, if(ispseudoprime(2*k*p-1), return(k))) \\ Felix Fröhlich, Apr 17 2019

a(n) = A053989(A000040(n))/2 for n <> 3. - Robert Israel, Apr 18 2019

cp's OEIS Frontend

A181715 Length of the complete Cunningham chain of the second kind starting with prime(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A278229 Least number with the prime signature of 2*prime(n) - 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Scheme

Formula

A285706 a(n) = number of iterations x -> A064216(x) needed to reach a nonprime number when starting from prime(n), a(1) = a(2) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Scheme

Formula

A023583 Greatest prime divisor of 2*prime(n)-1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A307368 a(n) is the minimal positive integer such that 2*a(n)*prime(n)-1 equals another prime.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A307368 a(n) is the minimal positive integer such that 2a(n)prime(n)-1 equals another prime.