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

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

1,4

Comments

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

Links

Table of n, a(n) for n=1..85.

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.

Example

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.

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 *)

Crossrefs

Cf. A027750, A092506, A243865.

Sequence in context: A001442 A226952 A158285 * A334112 A355625 A286238

Adjacent sequences: A277991 A277992 A277993 * A277995 A277996 A277997

Keyword

nonn

Author

Juri-Stepan Gerasimov, Nov 07 2016

Extensions

Corrected by Robert Israel, Nov 08 2016

Status

approved