A368216 Number of divisors of n that are antiharmonic numbers (A020487).

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 3, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 3, 2, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 1, 2

Offset: 1

Author

Marius A. Burtea, Jan 15 2024

Keywords

• nonn

Comments

Differs from A046951 for n = 20, 40, 50, 60, 80, ....

Examples

			a(1) = 1 because 1 has only one divisor 1 = A020487(1) antiharmonic number.
a(4) = 2 because 4 has divisors 1 = A020487(1) and 4 = A020487(2), antiharmonic numbers.
		

Crossrefs

Cf. A005117, A020487, A046951, A368215.

Programs

Magma

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

Mathematica

a[n_] := DivisorSum[n, 1 &, Divisible[DivisorSigma[2, #], DivisorSigma[1, #]] &]; Array[a, 100] (* Amiram Eldar, Jan 21 2024 *)

Formula

a(p^k) = floor((k + 2)/2), p prime, k >= 1.

a(p*q) = 1, for p, q prime, p <> q.

a(A005117(k)) = 1, k >= 1.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} 1/A020487(k) = 1.784... . - Amiram Eldar, Jan 26 2024