cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

Views

Author

Labos Elemer, Aug 28 2001

Keywords

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

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

A094502 a(n) = A000203(A046528(n)): sigma of those numbers whose sigma is a power of 2, in order of appearance.

Original entry on oeis.org

1, 4, 8, 32, 32, 128, 128, 256, 512, 1024, 1024, 4096, 4096, 8192, 16384, 32768, 32768, 65536, 131072, 131072, 262144, 262144, 524288, 524288, 1048576, 1048576, 1048576, 2097152, 2097152, 4194304, 4194304, 4194304, 4194304, 8388608
Offset: 1

Views

Author

Labos Elemer, Jun 02 2004

Keywords

Comments

Observe that certain powers of 2 do not arise as sum of divisors of something: 2,16,64,2048. Are there more? Yes, see A094505 and A078426.

Crossrefs

Cf. A000203, A046528, A078426, A094505.

Programs

  • Mathematica
    {ta=Table[0, {100}], u=1}; Do[If[IntegerQ[Log[2, DivisorSigma[1, n]]], Print[n];ta[[u]]=n;u=u+1], {n, 1, 100000000}] DivisorSigma[1, ta]
  • PARI
    isok(n) = (n==1) || (ispower(sigma(n), , &r) && (r==2));
for(n=1, 1e7, if(isok(n), print1(sigma(n)", "))) \\ Altug Alkan, Nov 01 2015

Formula

a(n) = 2^A048947(n). - R. J. Mathar, Sep 22 2016

A295043 a(n) is the largest number k such that sigma(k) = 2^n or 0 if no such k exists.

Original entry on oeis.org

1, 0, 3, 7, 0, 31, 0, 127, 217, 381, 889, 0, 3937, 8191, 11811, 27559, 57337, 131071, 253921, 524287, 1040257, 1777447, 4063201, 7281799, 16646017, 32247967, 66584449, 116522119, 225735769, 516026527, 1073602561, 2147483647, 4294434817, 7515217927, 15032385529
Offset: 0

Views

Author

Jaroslav Krizek, Nov 13 2017

Keywords

Comments

If a(n) > 0, then it is a term of A046528 (numbers that are a product of distinct Mersenne primes).

Examples

			a(0) = 1 because 1 is the largest number k with sigma(k) = 1 = 2^0.
a(5) = 31 because 31 is the largest number k with sigma(k) = 32 = 2^5.
a(6) = 0 because there is no number k with sigma(k) = 64 = 2^6.

Links

Crossrefs

Cf. A000043, A000203, A046528, A048947, A057637, A180221.
Cf. A247956 (the smallest number k instead of the largest).
Cf. A078426 (no solution to the equation sigma(x)=2^n).
A000668 (Mersenne primes) is a subsequence.

Programs

  • PARI
    a(n) = {local(r, k); r=0; for(k=1, 2^n, if(sigma(k) == 2^n, r=k)); return(r)}; \\ Michael B. Porter, Nov 14 2017
  • PARI
    a(n) = forstep(k=2^n, 1, -1, if (sigma(k)==2^n, return (k))); return (0) \\ Rémy Sigrist, Jan 08 2018
  • PARI
    a(n) = invsigmaMax(1<Amiram Eldar, Dec 20 2024, using Max Alekseyev's invphi.gp

Formula

a(A078426(n)) = 0.
a(A180221(n)) > 0.
a(n) <= 2^n - 1 with equality when n is a Mersenne exponent (A000043). - Michael B. Porter, Nov 14 2017
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