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A063883
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Number of ways writing n as a sum of different Mersenne prime exponents (terms of A000043).
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5
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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
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OFFSET
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1,5
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COMMENTS
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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
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LINKS
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FORMULA
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EXAMPLE
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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.
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MAPLE
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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):
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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STATUS
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approved
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