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A063883 Number of ways writing n as a sum of different Mersenne prime exponents (terms of A000043). 4
0, 1, 1, 0, 2, 0, 2, 1, 1, 2, 0, 2, 1, 1, 2, 1, 2, 2, 2, 3, 2, 4, 2, 4, 3, 3, 4, 2, 4, 2, 4, 3, 3, 4, 3, 4, 4, 4, 5, 4, 5, 4, 4, 5, 3, 5, 3, 4, 4, 3, 5, 3, 5, 4, 4, 5, 4, 5, 4, 4, 5, 3, 5, 4, 3, 6, 2, 6, 3, 5, 5, 3, 6, 3, 5, 4, 4, 4, 4, 4, 4, 4, 5, 3, 6, 3, 5, 5, 4, 6, 3, 7, 3, 6, 5, 5, 6, 5, 6, 5, 6, 6, 5, 6, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

This sequence appears to be growing. However, for 704338 < n < 756839, a(n) is 0. See A078426 for the n such that a(n)=0. - T. D. Noe, Oct 12 2006

Numbers k such that sigma(k) = 2^n. - Juri-Stepan Gerasimov, Mar 08 2017

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = A054973(2^n). - Michel Marcus, Mar 08 2017

EXAMPLE

n = 50 = 2 + 5 + 7 + 17 + 19 = 2 + 17 + 31 = 19 + 31, so a(50) = 3. The first numbers for which the number of these Mersenne-exponent partitions is k = 0, 1, 2, 3, 4, 5, 6, 7, 8 are 1, 2, 5, 20, 22, 39, 66, 92, 107, respectively.

MAPLE

N:= 500: # to get the first N terms

G:= mul(1+x^i, i=select(t -> numtheory:-mersenne(t)::integer, [$1..N])):

S:= series(G, x, N+1):

seq(coeff(S, x, n), n=1..N); # Robert Israel, Sep 22 2016

MATHEMATICA

exponents[n_] := Reap[For[k = 1, k <= n, k++, If[PrimeQ[2^k-1], Sow[k]]]][[2, 1]]; r[n_] := Module[{ee, x, xx}, ee = exponents[n]; xx = Array[x, Length[ee]]; Reduce[And @@ (0 <= # <= 1 & /@ xx) && xx.ee == n, xx, Integers]]; a[n_] := Which[rn = r[n]; Head[rn] === Or, Length[rn],  Head[rn] === And, 1, Head[rn] === Equal, 1, rn === False, 0, True, Print["error ", rn]]; a[1] = 0; Table[a[n], {n, 1, 105}] (* Jean-Fran├žois Alcover, Feb 05 2014 *)

PROG

(PARI) first(lim)=my(M=[2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667], x='x); if(lim>M[#M], error("Need more Mersenne exponents to compute further")); M=select(p->p<=lim, M); Vec(prod(i=1, #M, 1+x^M[i], O(x^(lim\1+1))+1)) \\ Charles R Greathouse IV, Mar 08 2017

(PARI) a(n) = sum(k=1, 2^n+1, sigma(k)==2^n); \\ Michel Marcus, Mar 07 2017

CROSSREFS

Cf. A000043, A046528, A048947, A063889, A054784.

Numbers n such that a(n) = m: A078426 (m = 0), A283160 (m = 1).

Sequence in context: A295672 A024161 A035156 * A079691 A104450 A281460

Adjacent sequences:  A063880 A063881 A063882 * A063884 A063885 A063886

KEYWORD

nonn,nice

AUTHOR

Labos Elemer, Aug 28 2001

STATUS

approved

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Last modified February 17 15:19 EST 2019. Contains 320220 sequences. (Running on oeis4.)