A373968 a(n) is the number of divisors of n that are Duffinian numbers (A003624).

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 2, 1, 0, 0, 0, 4, 0, 0, 1, 3, 0, 0, 1, 2, 0, 1, 0, 1, 1, 0, 0, 3, 1, 2, 0, 1, 0, 2, 1, 2, 1, 0, 0, 1, 0, 0, 3, 5, 1, 0, 0, 1, 0, 1, 0, 4, 0, 0, 2, 1, 1, 1, 0, 3, 3, 0, 0, 2, 1, 0, 0, 2

Offset: 1

Author

Marius A. Burtea, Jul 12 2024

Keywords

• nonn

Examples

			Since A003624(1) = 4 then a(1) = a(2) = a(3) = 0 and a(4) = 1.
a(8) = 2 because 8 has the divisors 4 = A003624(1) and 8 = A003624(2).
		

Crossrefs

Cf. A000203, A003624, A009194.

Programs

Magma

f:=func; [#[d:d in Divisors(k)|f(d)]:k in [1..100]];

Mathematica

a[n_] := DivisorSum[n, 1 &, CompositeQ[#] && CoprimeQ[#, DivisorSigma[1, #]] &]; Array[a, 100] (* Amiram Eldar, Jul 19 2024 *)

Formula

a(p^k) = k - 1, for p prime and k >= 1.