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A046051
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Number of prime factors of Mersenne number M(n) = 2^n - 1 (counted with multiplicity).
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43
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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
(list;
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refs;
listen;
history;
text;
internal format)
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OFFSET
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1,4
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COMMENTS
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Length of row n of A001265.
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LINKS
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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
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FORMULA
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Mobius transform of A085021. - T. D. Noe, Jun 19 2003
a(n) = A001222(A000225(n)). - Michel Marcus, Jun 06 2019
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EXAMPLE
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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)
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MAPLE
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with(numtheory): P:=proc(n) local a, k; a:=ifactors(2^n-1)[2];
add(a[k][2], k=1..nops(a)); end: seq(P(i), i=1..99); # Paolo P. Lava, Jul 18 2018
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n)=bigomega(2^n-1) \\ Charles R Greathouse IV, Apr 01 2013
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CROSSREFS
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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.
Sequence in context: A319149 A344462 A321887 * A025812 A263001 A109698
Adjacent sequences: A046048 A046049 A046050 * A046052 A046053 A046054
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KEYWORD
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nonn
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AUTHOR
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Eric W. Weisstein
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STATUS
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approved
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