%I #66 Feb 08 2024 07:12:39

%S 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,

%T 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,

%U 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

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

%C Length of row n of A001265.

%H Sean A. Irvine, <a href="/A046051/b046051.txt">Table of n, a(n) for n = 1..1206</a> (terms 1..500 from T. D. Noe)

%H J. Brillhart et al., <a href="http://dx.doi.org/10.1090/conm/022">Factorizations of b^n +- 1</a>, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

%H Alex Kontorovich, Jeff Lagarias, <a href="https://arxiv.org/abs/1808.03235">On Toric Orbits in the Affine Sieve</a>, arXiv:1808.03235 [math.NT], 2018.

%H R. Mestrovic, <a href="http://arxiv.org/abs/1202.3670">Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof</a>, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018.

%H S. S. Wagstaff, Jr., <a href="http://www.cerias.purdue.edu/homes/ssw/cun/index.html">The Cunningham Project</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MersenneNumber.html">Mersenne Number</a>

%F Mobius transform of A085021. - _T. D. Noe_, Jun 19 2003

%F a(n) = A001222(A000225(n)). - _Michel Marcus_, Jun 06 2019

%e a(4) = 2 because 2^4 - 1 = 15 = 3*5.

%e From _Gus Wiseman_, Jul 04 2019: (Start)

%e The sequence of Mersenne numbers together with their prime indices begins:

%e 1: {}

%e 3: {2}

%e 7: {4}

%e 15: {2,3}

%e 31: {11}

%e 63: {2,2,4}

%e 127: {31}

%e 255: {2,3,7}

%e 511: {4,21}

%e 1023: {2,5,11}

%e 2047: {9,24}

%e 4095: {2,2,3,4,6}

%e 8191: {1028}

%e 16383: {2,14,31}

%e 32767: {4,11,36}

%e 65535: {2,3,7,55}

%e 131071: {12251}

%e 262143: {2,2,2,4,8,21}

%e 524287: {43390}

%e 1048575: {2,3,3,5,11,13}

%e (End)

%t a[q_] := Module[{x, n}, x=FactorInteger[2^n-1]; n=Length[x]; Sum[Table[x[i][2], {i, n}][j], {j, n}]]

%t a[n_Integer] := PrimeOmega[2^n - 1]; Table[a[n], {n,200}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 22 2011 *)

%o (PARI) a(n)=bigomega(2^n-1) \\ _Charles R Greathouse IV_, Apr 01 2013

%Y 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).

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

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

%K nonn

%O 1,4

%A _Eric W. Weisstein_