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

1,4

Comments

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

Links

Table of n, a(n) for n=1..88.

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

Example

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.

Mathematica

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

Prog

(Magma) f:=func<n|DivisorSigma(2, n) mod DivisorSigma(1, n) eq 0>; [#[d:d in Divisors(k)|f(d)]:k in [1..100]];

Crossrefs

Cf. A005117, A020487, A046951, A368215.

Sequence in context: A299090 A046951 A159631 * A335428 A368781 A050377

Adjacent sequences: A368213 A368214 A368215 * A368217 A368218 A368219

Keyword

nonn

Author

Marius A. Burtea, Jan 15 2024

Status

approved