|
%I #17 Jan 26 2024 05:09:39
%S 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,
%T 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,
%U 1,1,1,4,1,1,2,2,1,1,1,4,3,1,1,2,1,1,1,2
%N Number of divisors of n that are antiharmonic numbers (A020487).
%C Differs from A046951 for n = 20, 40, 50, 60, 80, ....
%F a(p^k) = floor((k + 2)/2), p prime, k >= 1.
%F a(p*q) = 1, for p, q prime, p <> q.
%F a(A005117(k)) = 1, k >= 1.
%F 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
%e a(1) = 1 because 1 has only one divisor 1 = A020487(1) antiharmonic number.
%e a(4) = 2 because 4 has divisors 1 = A020487(1) and 4 = A020487(2), antiharmonic numbers.
%t a[n_] := DivisorSum[n, 1 &, Divisible[DivisorSigma[2, #], DivisorSigma[1, #]] &]; Array[a, 100] (* _Amiram Eldar_, Jan 21 2024 *)
%o (Magma) f:=func<n|DivisorSigma(2,n) mod DivisorSigma(1,n) eq 0>; [#[d:d in Divisors(k)|f(d)]:k in [1..100]];
%Y Cf. A005117, A020487, A046951, A368215.
%K nonn
%O 1,4
%A _Marius A. Burtea_, Jan 15 2024
|
