This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A380399 #10 Jan 25 2025 08:29:52
%S A380399 0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,3,0,0,
%T A380399 0,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,4,0,0,0,0,
%U A380399 0,0,0,2,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0
%N A380399 The number of nonunitary divisors of n that are perfect powers (A001597).
%H A380399 Amiram Eldar, <a href="/A380399/b380399.txt">Table of n, a(n) for n = 1..10000</a>
%F A380399 a(n) = Sum_{d|n, gcd(d, n/d) > 1} [d in A001597], where [] is the Iverson bracket.
%F A380399 a(n) = A091050(n) - A380398(n).
%F A380399 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A072102 + Sum_{k>=2} mu(k)*(zeta(k)/zeta(k+1) - 1) = Sum_{k>=2} mu(k)*zeta(k)*(1/zeta(k+1)-1) = 0.38105110303589889319..., where mu is the Moebius function (A008683).
%e A380399 a(16) = 2 since 16 have 2 nonunitary divisors that are perfect powers, 4 = 2^2 and 8 = 2^3.
%e A380399 a(32) = 3 since 32 have 3 nonunitary divisors that are perfect powers, 4 = 2^3, 8 = 2^3, and 16 = 2^4.
%t A380399 ppQ[n_] := n == 1 || GCD @@ FactorInteger[n][[;; , 2]] > 1; a[n_] := DivisorSum[n, 1 &, !CoprimeQ[#, n/#] && ppQ[#] &]; Array[a, 100]
%o A380399 (PARI) a(n) = sumdiv(n, d, gcd(d, n/d) > 1 && (d == 1 || ispower(d)));
%Y A380399 Cf. A001597, A008683, A048105, A072102, A091050, A380398.
%K A380399 nonn,easy
%O A380399 1,16
%A A380399 _Amiram Eldar_, Jan 23 2025