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.

A277994 Number of unordered integer pairs of the form {k | n, (k + 2^m) | n}, where k >= 1, m >= 0.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 0, 3, 2, 3, 0, 8, 0, 1, 3, 4, 1, 6, 0, 6, 2, 1, 0, 12, 1, 1, 2, 2, 0, 9, 0, 5, 3, 3, 2, 11, 0, 1, 1, 9, 0, 7, 0, 2, 5, 1, 0, 16, 0, 3, 2, 2, 0, 6, 1, 4, 2, 1, 0, 17, 0, 1, 4, 6, 3, 8, 0, 5, 1, 5, 0, 17, 0, 1, 3, 2, 1, 4, 0, 12, 2, 1, 0, 13, 2
Offset: 1

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Author

Juri-Stepan Gerasimov, Nov 07 2016

Keywords

Comments

Number of power-two-difference-divisor pairs of n.

Examples

			The positive divisors of 10 are 1, 2, 5, 10. Of these, {1 | 10, (1 + 2^0) | 10} = {1, 2}, {1 | 10, (1 + 2^2) | 10} = {1, 5}, {2 | 10, (2 + 2^3) | 10} = {2, 10}. So a(10) = 3.
		

Crossrefs

Programs

  • Maple
    f:=proc(n) local D,k;
      D:= numtheory:-divisors(n);
      add(nops(D intersect map(`+`,D,2^k)), k=0..ilog2(n-1));
    end proc:
    map(f, [$1..100]); # Robert Israel, Nov 08 2016
  • Mathematica
    f[n_] := Module[{dd = Divisors[n], k}, Sum[Length[dd ~Intersection~ (dd + 2^k)], {k, 0, Log[2, n - 1]}]];
    Array[f, 100] (* Jean-François Alcover, Jul 29 2020, after Robert Israel *)

Formula

Dirichlet g.f.: zeta(s) Sum_{k>=0} Sum_{m>=1} 1/lcm(m, m+2^k)^s. - Robert Israel, Nov 08 2016
a(2^n) = n, a(A092506(n)) = 1.

Extensions

Corrected by Robert Israel, Nov 08 2016