A003260 -id:A003260 - cp's OEIS frontend

A054992 Number of prime factors of 2^n + 1 (counted with multiplicity).

1, 1, 2, 1, 2, 2, 2, 1, 4, 3, 2, 2, 2, 3, 4, 1, 2, 4, 2, 2, 4, 3, 2, 3, 4, 4, 6, 2, 3, 6, 2, 2, 5, 4, 5, 4, 3, 4, 4, 2, 3, 6, 2, 3, 7, 5, 3, 3, 3, 7, 6, 3, 3, 6, 6, 3, 5, 3, 4, 4, 2, 5, 7, 2, 6, 6, 3, 4, 5, 7, 3, 5, 3, 5, 7, 4, 6, 10, 2, 3, 10, 5, 6, 5, 4, 5, 5, 4, 4, 11, 6, 2, 5, 4, 5, 3, 5, 6, 9, 6, 2, 9, 3

Offset: 1

Arne Ring (arne.ring(AT)epost.de), May 30 2000

nonn

The length of row n in A001269.

			a(3) = 2 because 2^3 + 1 = 9 = 3*3.

Max Alekseyev, Table of n, a(n) for n = 1..1128
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n+1): A057934 (b=10), A057935 (b=9), A057936 (b=8), A057937 (b=7), A057938 (b=6), A057939 (b=5), A057940 (b=4), A057941 (b=3), this sequence (b=2).
Cf. A000051, A002586, A002587, A003260, A001222, A001269, A001348, A054988, A054989, A054990, A054991, A000978.
Cf. A046051 (number of prime factors of 2^n-1).
Cf. A086257 (number of primitive prime factors).

a[n_] := Module[{x=FactorInteger[2^n+1]}, Sum[x[[i]][[2]], {i, Length[x]}]]
A054992[n_Integer] := PrimeOmega[2^n + 1]; Table[A054992[n], {n,103}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)

a(n)=bigomega(2^n+1) \\ Charles R Greathouse IV, Apr 29 2015

a(n) = A046051(2n) - A046051(n). - T. D. Noe, Jun 18 2003

a(n) = A001222(A000051(n)). - Amiram Eldar, Oct 04 2019

Extended by Patrick De Geest, Oct 01 2000
Terms to a(500) in b-file from T. D. Noe, Nov 10 2007
Deleted duplicate (and broken) Wagstaff link. - N. J. A. Sloane, Jan 18 2019
a(500)-a(1062) in b-file from Amiram Eldar, Oct 04 2019
a(1063)-a(1128) in b-file from Max Alekseyev, Jul 15 2023, Mar 15 2025

A016047 Smallest prime factor of Mersenne numbers 2^p-1, where p is prime.

Original entry on oeis.org

3, 7, 31, 127, 23, 8191, 131071, 524287, 47, 233, 2147483647, 223, 13367, 431, 2351, 6361, 179951, 2305843009213693951, 193707721, 228479, 439, 2687, 167, 618970019642690137449562111, 11447, 7432339208719, 2550183799, 162259276829213363391578010288127

Offset: 1

Robert G. Wilson v

"Mersenne numbers", here, means A001348. Compare to sequence A049479, where "Mersenne numbers" is used in the sense of A000225. - Lekraj Beedassy, Jun 11 2009
Submitted new b-file withdrawing last three terms previously submitted.  I had, before submitting that b-file, checked that the smallest known factors of incompletely factored Mersenne numbers was less than the known trial factoring limits recorded by Will Edgington in his LowM.txt file which can be found on his Mersenne page, (see link above.)  I have now discovered that I inadvertently omitted the purported a(203) from that check. - Daran Gill, Apr 05 2013
The would-be a(203) corresponds to 2^1237-1 which is currently P70*C303. Trial factoring has only been done to 60 bits, and since its difficulty doubles whenever the bit length is incremented by one, it cannot exhaustively search the space smaller than the sole known 70-digit (231-bit) factor. Probabilistic ECM testing indicates only that it is extremely unlikely that there is any undiscovered factor with digit-size smaller than the high fifties. See GIMPS links. - Gord Palameta, Aug 16 2018

Daran Gill, Table of n, a(n) for n = 1..202 (first 95 terms from T. D. Noe)
C. K. Caldwell, Mersenne Primes
Will Edgington, Mersenne Page
GIMPS, PrimeNet Known Factors of Mersenne Numbers (discovered recently for p < 2000), Exponent status of 2^1237 - 1, ECM testing status of 2^1237 - 1
Brady Haran and Matt Parker, How they found the World's Biggest Prime Number - Numberphile, Numberphile video (2016).

Cf. A000668 (a subsequence), A003260, A001348, A020639, A046800.

a:= n-> min(numtheory[factorset](2^ithprime(n)-1)):
seq(a(n), n=1..28);  # Alois P. Heinz, Oct 01 2024

a = {}; Do[If[PrimeQ[n], w = 2^n - 1; c = FactorInteger[w]; b = c[[1]][[1]]; AppendTo[a, b]], {n, 2, 100}]; a (* Artur Jasinski, Dec 11 2007 *)

forprime(p=2,150,print1(factor(2^p-1)[1,1],", "))

a(n) = A020639(A001348(n)). - Alois P. Heinz, Oct 01 2024

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

Original entry on oeis.org

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.

A088863 Number of prime factors of n-th Mersenne number M(p_n).

Original entry on oeis.org

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

Offset: 1

Jeppe Stig Nielsen, Nov 25 2003

nonn

			a(5)=2 because M(p_5)=M(11)=2047 has 2 (not necessarily distinct) prime factors.

Max Alekseyev, Table of n, a(n) for n = 1..197 (first 137 terms from Herman Jamke)
Herman Jamke, The first 137 terms in detail

Cf. A001348, A003260, A046051, A046932.

seq(nops(ifactor(2^ithprime(n)-1)),n=1..32); # Emeric Deutsch, Dec 23 2004

Do[m = 2^Prime[n] - 1; Print[Plus @@ Last /@ FactorInteger[m]], {n, 1, 50}] (* Ryan Propper, Jul 31 2005 *)

for(n=1,137,print1(bigomega(2^prime(n)-1)",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 28 2007

a(n) = A001222(A001348(n)) = A001222(A000225(A000040(n))).

14 more terms from Emeric Deutsch, Dec 23 2004
More terms from Ryan Propper, Jul 31 2005
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 28 2007

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

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

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

A236128 Primes p such that gpf(gpf(2^p-1)-1) = p.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 29, 53

Offset: 1

Thomas Ordowski, Jan 19 2014

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 20 2014
Conjecture: gpf(gpf(2^p-1)-1) = p for finitely many p.
Conjecture: gpf(lpf(2^p-1)-1) = p for infinitely many p.
Namely, for p = 2, 3, 5, 7, 11, 13, 23, 29, 37, 43, 47, 53, ... - Michael B. Porter, Jan 26 2014
Note that gpf(lpf(2^p-1)-1) = gpf(gpf(2^p-1)-1) = p for p = 2, 3, 5, 7, 11, 13, 29, 53. See DATA.

			For prime p=2, 2^p-1=3, gpf(3)=3, gpf(3-1)=2, so 2 is in the sequence.
For prime p=3, 2^p-1=7, gpf(7)=7, gpf(7-1)=3, so 3 is in the sequence.

GIMPS, Exponent Status

Cf. A003260, A006530.

Select[Prime[Range[25]], FactorInteger[FactorInteger[2^# - 1][[-1, 1]] - 1][[-1, 1]] == # &] (* Alonso del Arte, Jan 19 2014 *)

isok(p) = isprime(p) && (q = (vecmax(factor(2^p-1)[,1]))) && (vecmax(factor(q-1)[,1]) == p); \\ Michel Marcus, Jan 19 2014

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

Original entry on oeis.org

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

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

cp's OEIS Frontend

A054992 Number of prime factors of 2^n + 1 (counted with multiplicity).

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A016047 Smallest prime factor of Mersenne numbers 2^p-1, where p is prime.

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

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Magma

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A088863 Number of prime factors of n-th Mersenne number M(p_n).

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A136031 Largest prime factor of composite 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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A136033 a(n) = smallest number k such that number of prime factors of 2^k-1 is exactly n (counted with multiplicity).

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