%I #15 Jul 29 2020 03:17:26

%S 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,

%T 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,

%U 5,1,5,0,17,0,1,3,2,1,4,0,12,2,1,0,13,2

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

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

%F Dirichlet g.f.: zeta(s) Sum_{k>=0} Sum_{m>=1} 1/lcm(m, m+2^k)^s. - _Robert Israel_, Nov 08 2016

%F a(2^n) = n, a(A092506(n)) = 1.

%e 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.

%p f:=proc(n) local D,k;

%p D:= numtheory:-divisors(n);

%p add(nops(D intersect map(`+`,D,2^k)), k=0..ilog2(n-1));

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Nov 08 2016

%t f[n_] := Module[{dd = Divisors[n], k}, Sum[Length[dd ~Intersection~ (dd + 2^k)], {k, 0, Log[2, n - 1]}]];

%t Array[f, 100] (* _Jean-François Alcover_, Jul 29 2020, after _Robert Israel_ *)

%Y Cf. A027750, A092506, A243865.

%K nonn

%O 1,4

%A _Juri-Stepan Gerasimov_, Nov 07 2016

%E Corrected by _Robert Israel_, Nov 08 2016