A046051 Number of prime factors of Mersenne number M(n) = 2^n - 1 (counted with multiplicity).

0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 2, 5, 1, 3, 3, 4, 1, 6, 1, 6, 4, 4, 2, 7, 3, 3, 3, 6, 3, 7, 1, 5, 4, 3, 4, 10, 2, 3, 4, 8, 2, 8, 3, 7, 6, 4, 3, 10, 2, 7, 5, 7, 3, 9, 6, 8, 4, 6, 2, 13, 1, 3, 7, 7, 3, 9, 2, 7, 4, 9, 3, 14, 3, 5, 7, 7, 4, 8, 3, 10, 6, 5, 2, 14, 3, 5, 6, 10, 1, 13, 5, 9, 3, 6, 5, 13, 2, 5, 8

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn

Comments

Length of row n of A001265.

Examples

			a(4) = 2 because 2^4 - 1 = 15 = 3*5.
From _Gus Wiseman_, Jul 04 2019: (Start)
The sequence of Mersenne numbers together with their prime indices begins:
        1: {}
        3: {2}
        7: {4}
       15: {2,3}
       31: {11}
       63: {2,2,4}
      127: {31}
      255: {2,3,7}
      511: {4,21}
     1023: {2,5,11}
     2047: {9,24}
     4095: {2,2,3,4,6}
     8191: {1028}
    16383: {2,14,31}
    32767: {4,11,36}
    65535: {2,3,7,55}
   131071: {12251}
   262143: {2,2,2,4,8,21}
   524287: {43390}
  1048575: {2,3,3,5,11,13}
(End)
		

Links

• Sean A. Irvine, Table of n, a(n) for n = 1..1206 (terms 1..500 from T. D. Noe)
• J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
• Alex Kontorovich, Jeff Lagarias, On Toric Orbits in the Affine Sieve, arXiv:1808.03235 [math.NT], 2018.
• R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018.
• S. S. Wagstaff, Jr., The Cunningham Project
• Eric Weisstein's World of Mathematics, Mersenne Number

Crossrefs

bigomega(b^n-1): A057951 (b=10), A057952 (b=9), A057953 (b=8), A057954 (b=7), A057955 (b=6), A057956 (b=5), A057957 (b=4), A057958 (b=3), this sequence (b=2).

Cf. A000043, A000668, A001348, A054988, A054989, A054990, A054991, A054992, A085021.

Cf. A000225, A001221, A001222, A046800, A049093, A059305, A325610, A325611, A325612, A325625.

Programs

Mathematica

a[q_] := Module[{x, n}, x=FactorInteger[2^n-1]; n=Length[x]; Sum[Table[x[i][2], {i, n}][j], {j, n}]]
a[n_Integer] := PrimeOmega[2^n - 1]; Table[a[n], {n,200}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)

PARI

a(n)=bigomega(2^n-1) \\ Charles R Greathouse IV, Apr 01 2013

Formula

Mobius transform of A085021. - T. D. Noe, Jun 19 2003

a(n) = A001222(A000225(n)). - Michel Marcus, Jun 06 2019