A016047 -id:A016047 - cp's OEIS frontend

A122094 Prime divisors of Mersenne numbers. Primes p such that the multiplicative order of 2 modulo p is prime.

3, 7, 23, 31, 47, 89, 127, 167, 223, 233, 263, 359, 383, 431, 439, 479, 503, 719, 839, 863, 887, 983, 1103, 1319, 1367, 1399, 1433, 1439, 1487, 1823, 1913, 2039, 2063, 2089, 2207, 2351, 2383, 2447, 2687, 2767, 2879, 2903, 2999, 3023, 3119, 3167, 3343

Offset: 1

Max Alekseyev, Oct 25 2006

nonn

Except for the first term (3), all terms are 1 or 7 (mod 8). - William Hu, May 03 2024

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A001348, A016047, A003260, A000668, A137332.

Cf. A089162 (this list sorted by q).

[p: p in PrimesInInterval(2, 4000) | IsPrime(Modorder(2, p))]; // Vincenzo Librandi, Oct 28 2016

Reap[For[p=2, p<10^5, p=NextPrime[p], If[PrimeQ[MultiplicativeOrder[2, p]], Sow[p]]]][[2, 1]] (* Jean-François Alcover, Dec 10 2015 *)
Select[Prime@ Range@ 500, PrimeQ@ MultiplicativeOrder[2, #] &] (* Michael De Vlieger, Oct 28 2016 *)

forprime(p=3,10^5,if(isprime(znorder(Mod(2,p))),print1(p,",")))

p is a prime divisor of a Mersenne number 2^q - 1 iff prime q is the multiplicative order of 2 modulo p.

A003260 Largest prime factor of n-th Mersenne number (A001348(n)).

Original entry on oeis.org

3, 7, 31, 127, 89, 8191, 131071, 524287, 178481, 2089, 2147483647, 616318177, 164511353, 2099863, 13264529, 20394401, 3203431780337, 2305843009213693951

Offset: 1

N. J. A. Sloane

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Michel Marcus and Gord Palameta, Table of n, a(n) for n = 1..197 (first 95 terms from T. D. Noe, derived from Brillhart et al.)
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
C. K. Caldwell, Mersenne primes
P. Erdős and T. N. Shorey, On the greatest prime factor of 2^p-1 for a prime p and other expressions, Acta Arith. 30:3 (1976), pp. 257-265.
C. L. Stewart, The greatest prime factor of a^n - b^n, Acta Arith. 26 (1975), pp. 427-433.
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A000668 (a subsequence), A001348, A016047, A046800.

a[n_] := FactorInteger[ 2^Prime[n] - 1 ][[-1, 1]]; Table[ a[n], {n, 1, 18}] (* Jean-François Alcover, Dec 20 2011 *)

a(n)=my(f=factor(2^prime(n)-1)[,1]);f[#f] \\ Charles R Greathouse IV, Dec 05 2012

Let p = prime(n). Erdős & Shorey show that a(n) >= kp log p for some effectively computable k >= 1. (Presumably k can be chosen as 7/log 27.) - Charles R Greathouse IV, Dec 05 2012

A136031 Largest prime factor of composite Mersenne numbers.

Original entry on oeis.org

89, 178481, 2089, 616318177, 164511353, 2099863, 13264529, 20394401, 3203431780337, 761838257287, 212885833, 9361973132609, 1113491139767, 57912614113275649087721, 13842607235828485645766393, 341117531003194129, 3976656429941438590393

Offset: 1

Artur Jasinski, Dec 11 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..183

Cf. A049479, A016047, A003260, A136030.

FactorInteger[2^#-1][[-1,1]]&/@Select[Prime[Range[40]],!PrimeQ[2^#-1]&] (* Harvey P. Dale, May 05 2012 *)
With[{c=Complement[Prime[Range[PrimePi[200]]],MersennePrimeExponent[ Range[ 15]]]},Table[ FactorInteger[ 2^p-1][[-1,1]],{p,c}]] (* Harvey P. Dale, Sep 06 2021 *)

a(n) >= A089158(n). - R. J. Mathar, May 01 2008

Corrected by Harvey P. Dale, May 05 2012

A136030 Smallest prime factor of composite Mersenne numbers.

Original entry on oeis.org

23, 47, 233, 223, 13367, 431, 2351, 6361, 179951, 193707721, 228479, 439, 2687, 167, 11447, 7432339208719, 2550183799, 745988807, 3391, 263, 32032215596496435569, 5625767248687, 86656268566282183151, 18121, 852133201, 150287, 2349023, 730753, 359, 43441, 383

Offset: 1

Artur Jasinski, Dec 11 2007

nonn

Max Alekseyev, Table of n, a(n) for n = 1..188

Cf. A049479, A016047.

a = {}; Do[If[PrimeQ[n] && !PrimeQ[2^n - 1], w = 2^n - 1; c = FactorInteger[w]; b = c[[1]][[1]]; AppendTo[a, b]], {n, 2, 130}]; a

lista() = {vi = readvec("c:/gp/bfiles/b054723.txt"); vm = vector(#vi, i, 2^vi[i]-1); for (i=1, #vm, print1(factor(vm[i])[1, 1], ", "););} \\ Michel Marcus, May 14 2014

More terms from Michel Marcus, May 14 2014
Terms to a(150) in b-file from Charles R Greathouse IV, May 14 2014
a(151)-a(188) in b-file added at the suggestion of Eric Chen by Max Alekseyev, Apr 25 2022

A089158 Second prime factor, if it exists, of Mersenne numbers.

Original entry on oeis.org

89, 178481, 1103, 616318177, 164511353, 9719, 4513, 69431, 3203431780337, 761838257287, 48544121, 2298041, 202029703, 57912614113275649087721, 13842607235828485645766393, 341117531003194129, 3976656429941438590393

Offset: 1

Cino Hilliard, Dec 06 2003

nonn

			The 5th Mersenne number 2^11 - 1 = 23*89 and 89 is the second prime divisor.
The 9th Mersenne number 2^23 - 1 = 47*178481 and 178481 is the second prime divisor.
Notice 23, 89 congruent to 1 mod 11 and 47, 178481 congruent to 1 mod 23.

Amiram Eldar, Table of n, a(n) for n = 1..183
Chris Caldwell, Mersenne Primes: History, Theorems and Lists.

Cf. A001348, A016047, A065341, A089159, A089992, A135978.

mersenne(b,n,d) = { c=0; forprime(x=2,n, c++; y = b^x-1; f=factor(y); v=component(f,1); ln = length(v); if(ln>=d,print1(v[d]",")); ) }

A089162 Triangle read by rows formed by the prime factors of Mersenne number 2^prime(n) - 1, n >= 1.

Original entry on oeis.org

3, 7, 31, 127, 23, 89, 8191, 131071, 524287, 47, 178481, 233, 1103, 2089, 2147483647, 223, 616318177, 13367, 164511353, 431, 9719, 2099863, 2351, 4513, 13264529, 6361, 69431, 20394401, 179951, 3203431780337, 2305843009213693951, 193707721, 761838257287

Offset: 1

Cino Hilliard, Dec 06 2003

All factors of Mersenne numbers 2^p - 1, where p is prime, are == 1 (mod p). See the first Caldwell link for a proof of the statement that if q divides M_p = 2^p-1 then q = 2kp + 1 for some integer k. - Comment corrected by Jonathan Sondow, Dec 29 2016

			The 16th Mersenne number 2^53-1 has the three prime factors 6361, 69431, 20394401.
See tail end of second row in the sequence. Each factor is == 1 (mod 53).
Triangle begins:
  3;
  7;
  31;
  127;
  23, 89;
  8191;
  131071;
  524287;
  47, 178481;
  233, 1103, 2089;
  2147483647;
  223, 616318177;
  13367, 164511353;
  431, 9719, 2099863;
  2351, 4513, 13264529;
  6361, 69431, 20394401;

Max Alekseyev, Rows n = 1..197, flattened (rows 1..167 from Jens Kruse Andersen)
R. P. Brent, New factors of Mersenne numbers.
Chris K. Caldwell, Mersenne Primes: History, Theorems and Lists.
Chris K. Caldwell, The Prime Glossary, Mersenne divisor.
Sam Wagstaff, The Cunningham Project.

Cf. A001348, A003260, A016047, A088863.

Cf. A122094 (sorted version of this list).

row[n_]:=First/@FactorInteger[2^Prime[n]-1]; Array[row,19]//Flatten (* Stefano Spezia, May 03 2024 *)

mersenne(b,n,d) = { c=0; forprime(x=2,n, c++; y = b^x-1; f=factor(y); v=component(f,1); ln = length(v); if(ln>=d,print1(v[d]",")); ) }

Definition corrected by Max Alekseyev, Jul 25 2023

A263686 Smallest prime factor of double Mersenne numbers.

Original entry on oeis.org

7, 127, 2147483647, 170141183460469231731687303715884105727, 338193759479, 231733529, 62914441, 295257526626031

Offset: 1

Arkadiusz Wesolowski, Oct 23 2015

A double Mersenne number is a Mersenne number of the form 2^(2^p - 1) - 1, where p is a Mersenne exponent (A000043).
From M. F. Hasler, Feb 28 2025: (Start)
The prime factors of Mersenne numbers 2^q - 1 must be of the form 2*q*k + 1.
The four smallest double Mersenne numbers (p = 2, 3, 5, 7 => q = 3, 7, 31, 127) are prime, so their smallest prime factor is equal to themselves, a(n) = M(q). This is equivalent to k = (2^(q-1)-1)/q, which is almost as large as M(q) itself: k = 1, 9 and 34636833 for the first three terms, and for q = 127, k has just three digits less than M(q) = a(4) itself. The prime p = 11 is not a Mersenne exponent.
The fifth term, a(5) = 2*(2^13-1)*k + 1 with k = 20644229 (which is prime) is the first proper divisor of the respective M(q), as are the next three, corresponding to p = 17, 19 and 31.
For p = 61, M(q) has 694127911065419642 digits, and so far no factor is known, but it is known that it has no factor less than 10^36. (End)

Double Mersennes Prime Search, History
Wikipedia, Double Mersenne number

Cf. A000043, A000668, A001348, A020639, A049479, A077586, A122094. Subsequence of A016047. Subsequence of A309130.

forprime(p=2,,q=2^p-1; !ispseudoprime(q) && next(); if(ispseudoprime(2^q-1), print1(2^q-1,", ");next()); forstep(r=2*q+1,+oo,2*q, !ispseudoprime(r) && next(); if(Mod(2,r)^q-1 == 0, print1(r,", ");next(2)))) \\ Jeppe Stig Nielsen, Aug 28 2019

a(n) = spf(MM(A000043(n))) = A049479(A000668(n)), where spf = A020639 is the smallest prime factor, A049479 = spf o M, M(p) = 2^p-1 = A000225(p), MM = M o M = A077585, A000668(n) = M(A000043(n)), A000043 are the Mersenne prime exponents. - M. F. Hasler, Mar 01 2025

A291691 Primes p such that gpf(lpf(2^p - 1) - 1) = p.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 23, 29, 37, 43, 47, 53, 73, 79, 83, 97, 113, 131, 151, 173, 179, 181, 191, 197, 211, 223, 233, 239, 251, 263, 277, 281, 283, 307, 317, 337, 353, 359, 367, 383, 397, 419, 431, 439, 443, 457, 461, 463, 467, 487, 491, 499

Offset: 1

Thomas Ordowski, Aug 30 2017

nonn

This sequence has not been proved to be infinite.
The terms p such that 2^p - 1 is a Mersenne prime are 2, 3, 5, 7, and 13.
If p is prime, then gpf(lpf(2^p - 1) - 1) >= p.
Primes q such that gpf(lpf(2^q - 1) - 1) > q are A292237.

			We have gpf(lpf(2^11 - 1) - 1) = gpf(23 - 1) = 11, so 11 is a term.

Charles R Greathouse IV, Table of n, a(n) for n = 1..119

Cf. A000040, A000043, A006530, A016047, A020639, A236128.

Contains A002515.

lpf[n_] := FactorInteger[n][[1, 1]]; gpf[n_] := FactorInteger[n][[-1, 1]]; Select[ Prime@ Range@ 45, gpf[lpf[2^# - 1] - 1] == # &] (* Giovanni Resta, Aug 30 2017 *)

listp(nn) = forprime(p=2, nn, if (vecmax(factor(vecmin(factor(2^p-1)[,1])-1)[,1]) == p, print1(p, ", "));); \\ Michel Marcus, Aug 30 2017

a(17)-a(26) from Michel Marcus, Aug 30 2017
a(27)-a(34) from Giovanni Resta, Aug 30 2017
a(35)-a(52) from Charles R Greathouse IV, Aug 30 2017

A089159 If Mersenne numbers have 3 or more factors, then list the third factor.

Original entry on oeis.org

2089, 2099863, 13264529, 20394401, 212885833, 9361973132609, 1113491139767, 65993, 165799, 1654058017289, 110211473, 70084436712553223, 1489459109360039866456940197095433721664951999121, 7648337, 39940132241, 14732265321145317331353282383

Offset: 1

Cino Hilliard, Dec 06 2003, corrected Nov 16 2006

nonn

			The 10th Mersenne number 2^29 - 1 = 233*1103*2089 and 2089 is the third prime factor. Notice these factors are congruent to 1 (mod 29).

Amiram Eldar, Table of n, a(n) for n = 1..144
Chris Caldwell, Mersenne Primes: History, Theorems and Lists.

Cf. A000043, A001348, A016047, A089158, A135977, A344515.

mersenne2(n) = { c=0; forprime(x=2, n, c++; y = 2^x-1; f=ifactor(y); if(length(f)>=3, print1(f[3]","); ) ) }
ifactor(n) = { local(f,j,k,flist); flist=[]; f=Vec(factor(n)); for(j=1,length(f[1]), for(k = 1,f[2][j],flist = concat(flist,f[1][j]) ); ); return(flist) }

A Mersenne number (A001348) is a number of the form 2^p - 1 where p is prime.

a(16) from Amiram Eldar, Jul 11 2024

A136033 a(n) = smallest number k such that number of prime factors of 2^k-1 is exactly n (counted with multiplicity).

Original entry on oeis.org

2, 4, 6, 16, 12, 18, 24, 40, 54, 36, 102, 110, 60, 72, 108, 140, 120, 156, 144, 200, 216, 210, 240, 180, 456, 288, 336, 300, 396, 480, 882, 360, 468, 700

Offset: 1

Artur Jasinski, Dec 11 2007

Cf. A000225, A003260, A016047, A046051, A049479, A088863, A136030, A136031.

N:= 24: # to get a(1) to a(N)
unknown:= N:
for k from 2 while unknown > 0 do
  q:= numtheory:-bigomega(2^k-1);
  if q <= N and not assigned(A[q]) then
     A[q]:= k;
     unknown:= unknown - 1;
  fi
od:
seq(A[i],i=1..N); # Robert Israel, Oct 24 2014

Module[{nn=250,tbl},tbl=Table[{k,PrimeOmega[2^k-1]},{k,nn}];Table[SelectFirst[tbl,#[[2]]==n&],{n,24}]][[;;,1]] (* The program generates the first 24 terms of the sequence. *)  (* Harvey P. Dale, May 25 2025 *)

a(n) = {k = 1; while(bigomega(2^k-1) != n, k++); k;} \\ Michel Marcus, Nov 04 2013

a(15)-a(20) from Michel Marcus, Nov 04 2013
a(21)-a(24) from Derek Orr, Oct 23 2014
a(25)-a(34) from Jinyuan Wang, Jun 07 2019

cp's OEIS Frontend

A122094 Prime divisors of Mersenne numbers. Primes p such that the multiplicative order of 2 modulo p is prime.

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A003260 Largest prime factor of n-th Mersenne number (A001348(n)).

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A136031 Largest prime factor of composite Mersenne numbers.

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A136030 Smallest prime factor of composite Mersenne numbers.

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A089158 Second prime factor, if it exists, of Mersenne numbers.

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A089162 Triangle read by rows formed by the prime factors of Mersenne number 2^prime(n) - 1, n >= 1.

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A263686 Smallest prime factor of double Mersenne numbers.

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A291691 Primes p such that gpf(lpf(2^p - 1) - 1) = p.

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