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

%I A063883 #40 Dec 23 2024 09:54:55
%S A063883 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,
%T A063883 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,
%U A063883 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
%N A063883 Number of ways writing n as a sum of different Mersenne prime exponents (terms of A000043).
%C A063883 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
%C A063883 Numbers k such that sigma(k) = 2^n. - _Juri-Stepan Gerasimov_, Mar 08 2017
%H A063883 T. D. Noe, <a href="/A063883/b063883.txt">Table of n, a(n) for n = 1..10000</a>
%F A063883 a(n) = A054973(2^n). - _Michel Marcus_, Mar 08 2017
%e A063883 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.
%p A063883 N:= 500: # to get the first N terms
%p A063883 G:= mul(1+x^i,i=select(t -> numtheory:-mersenne(t)::integer, [$1..N])):
%p A063883 S:= series(G,x,N+1):
%p A063883 seq(coeff(S,x,n),n=1..N); # _Robert Israel_, Sep 22 2016
%t A063883 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 *)
%t A063883 With[{e = MersennePrimeExponent[Range[10]]}, Rest@ CoefficientList[Product[1 + x^e[[i]], {i, 1, Length[e]}], x]] (* _Amiram Eldar_, Dec 23 2024 *)
%o A063883 (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
%o A063883 (PARI) a(n) = sum(k=1, 2^n+1, sigma(k)==2^n); \\ _Michel Marcus_, Mar 07 2017
%Y A063883 Cf. A000043, A046528, A048947, A054784, A054973, A063889.
%Y A063883 Numbers k such that a(k) = m: A078426 (m = 0), A283160 (m = 1).
%K A063883 nonn,nice
%O A063883 1,5
%A A063883 _Labos Elemer_, Aug 28 2001