A063883 Number of ways writing n as a sum of different Mersenne prime exponents (terms of A000043).

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

Offset: 1

Author

Labos Elemer, Aug 28 2001

Keywords

• nonn
• nice

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A000043, A046528, A048947, A054784, A054973, A063889.

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

Programs

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-François Alcover, Feb 05 2014 *)
With[{e = MersennePrimeExponent[Range[10]]}, Rest@ CoefficientList[Product[1 + x^e[[i]], {i, 1, Length[e]}], x]] (* Amiram Eldar, Dec 23 2024 *)

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

Formula

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