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
Keywords
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.
Programs
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Magma
f:=func
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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
Comments