A016047 -id:A016047 - cp's OEIS frontend

A244453 Prime factors of 2^A054723(n)-1, ordered by increasing n, then by increasing size of the factors.

23, 89, 47, 178481, 233, 1103, 2089, 223, 616318177, 13367, 164511353, 431, 9719, 2099863, 2351, 4513, 13264529, 6361, 69431, 20394401, 179951, 3203431780337, 193707721, 761838257287, 228479, 48544121, 212885833

Offset: 1

Felix Fröhlich, Jun 28 2014

Subsequence of A060443.

Prime factors of composite Mersenne numbers; A089162 with the Mersenne primes A000668 removed. - Jens Kruse Andersen, Jul 11 2014

			A054723(1) = 11. 2^11-1 = 2047 = 23*89. - _Jens Kruse Andersen_, Jul 11 2014
Triangle begins:
23, 89;
47, 178481;
233, 1103, 2089;
223, 616318177;
13367, 164511353;
431, 9719, 2099863;
2351, 4513, 13264529;
6361, 69431, 20394401;

Jens Kruse Andersen, Table of n, a(n) for n = 1..577
Sergey Nikitin, Euler-Fermat algorithm and some of its applications, 2018.
Sam Wagstaff, The Cunningham Project

Cf. A003260, A016047, A046800, A089162.

Map[FactorInteger, Select[2^Prime@Range@20 - 1, CompositeQ]][[All, All, 1]] // Flatten (* Michael De Vlieger, Nov 20 2018 *)

forprime(n=1, 100, m=2^n-1; if(!isprime(m), f=factor(m); for(i=1, #f~, print1(f[i,1]", ")))) \\ Jens Kruse Andersen, Jul 11 2014

A016048 Least k such that (2p_n)k + 1 | Mersenne(p_n), p_n = n-th prime, n >= 2.

Original entry on oeis.org

1, 3, 9, 1, 315, 3855, 13797, 1, 4, 34636833, 3, 163, 5, 25, 60, 1525, 18900352534538475, 1445580, 1609, 3, 17, 1, 3477359660913989536233495, 59, 36793758459, 12379533, 758220919762679268184943973309, 3421967, 15

Offset: 2

Robert G. Wilson v

nonn

M(p_n) = 2^p_n - 1 = (2*p_n)*j + 1 = [(2*p_n)*k_1 + 1] ... [(2*p_n)*k_i + 1], n >= 2 (i.e., odd prime p_n), i >= 1. Then k = Min(k_1, ..., k_i).

Chris K. Caldwell, Modular restrictions on Mersenne divisors

Cf. A000040, A016047.

a(n) = (A016047(n) - 1) / (2*A000040(n)). - Jeppe Stig Nielsen, Jul 18 2014

Definition edited, comment added by Daniel Forgues, Oct 06 2009

A136034 a(n) = smallest number k such that number of distinct prime factors of 2^k-1 is exactly n.

Original entry on oeis.org

1, 2, 4, 8, 12, 20, 24, 40, 36, 48, 88, 60, 72, 150, 132, 120, 156, 144, 200, 204, 210, 180, 324, 476, 288, 300, 432, 396, 480, 360, 468, 576, 700, 504, 420, 648, 540, 660, 792, 720

Offset: 0

Artur Jasinski, Dec 11 2007

First occurrence of n in A046800.

Cf. A000225, A003260, A016047, A046051, A046800, A049479, A088863, A136030, A136031, A136032, A136033.

With[{pn1=PrimeNu[2^Range[800]-1]},Table[Position[pn1,n,1,1],{n,0,40}]]//Flatten (* Harvey P. Dale, Jan 10 2025 *)

a(n) = my(k=1); while (omega(2^k-1) != n, k++); k; \\ Michel Marcus, Jan 09 2023

More terms from Julián Aguirre, Feb 04 2013
a(31)-a(39) from Chai Wah Wu, Oct 03 2019
a(0) = 1 inserted by Michel Marcus, Jan 09 2023

A236485 Primes p such that gpf(lpf(m)-1) = gpf(gpf(m)-1), where m = 2^p-1 is Mersenne composite.

Original entry on oeis.org

11, 29, 53

Offset: 1

Thomas Ordowski, Jan 27 2014

Conjecture: finitely many such p and gpf(lpf(m)-1) = gpf(gpf(m)-1) = p.

No more terms found up to p = 1277, 1277 being the first prime for which the complete factorization of 2^p-1 is not currently known (see GIMPS link). - Michel Marcus, Jan 29 2014

GIMPS, Exponent Status

Cf. A003260, A016047, A236128.

isok(p) = isprime(p) && (mc = (2^p-1)) && (mcpf = (factor(2^p-1))[, 1]) && (length(mcpf) > 1) && (gpf = vecmax(mcpf)) && (lpf = vecmin(mcpf)) && (vecmax(factor(gpf-1)[, 1]) == vecmax(factor(lpf-1)[, 1])); \\ Michel Marcus, Jan 27 2014

cp's OEIS Frontend

A244453 Prime factors of 2^A054723(n)-1, ordered by increasing n, then by increasing size of the factors.

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A016048 Least k such that (2p_n)k + 1 | Mersenne(p_n), p_n = n-th prime, n >= 2.

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A136034 a(n) = smallest number k such that number of distinct prime factors of 2^k-1 is exactly n.

Views

Author

Keywords

Comments

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Programs

Mathematica

PARI

Extensions

A236485 Primes p such that gpf(lpf(m)-1) = gpf(gpf(m)-1), where m = 2^p-1 is Mersenne composite.

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Author

Keywords

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PARI

cp's OEIS Frontend

A244453 Prime factors of 2^A054723(n)-1, ordered by increasing n, then by increasing size of the factors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A016048 Least k such that (2*p_n)*k + 1 | Mersenne(p_n), p_n = n-th prime, n >= 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A136034 a(n) = smallest number k such that number of distinct prime factors of 2^k-1 is exactly n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Extensions

A236485 Primes p such that gpf(lpf(m)-1) = gpf(gpf(m)-1), where m = 2^p-1 is Mersenne composite.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A016048 Least k such that (2p_n)k + 1 | Mersenne(p_n), p_n = n-th prime, n >= 2.