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A380398 The number of unitary divisors of n that are perfect powers (A001597).

%I A380398 #10 Jan 25 2025 08:29:39
%S A380398 1,1,1,2,1,1,1,2,2,1,1,2,1,1,1,2,1,2,1,2,1,1,1,2,2,1,2,2,1,1,1,2,1,1,
%T A380398 1,4,1,1,1,2,1,1,1,2,2,1,1,2,2,2,1,2,1,2,1,2,1,1,1,2,1,1,2,2,1,1,1,2,
%U A380398 1,1,1,3,1,1,2,2,1,1,1,2,2,1,1,2,1,1,1
%N A380398 The number of unitary divisors of n that are perfect powers (A001597).
%C A380398 First differs from A368978 at n = 32, from A007424 and A369163 at n = 36, from A278908, A307848, A358260 and A365549 at n = 64, and from A323308 at n = 72.
%C A380398 a(n) depends only on the prime signature of n (A118914).
%C A380398 The record values are 2^k, for k = 0, 1, 2, ..., and they are attained at A061742(k).
%C A380398 The sum of unitary divisors of n that are perfect powers is A380400(n).
%H A380398 Amiram Eldar, <a href="/A380398/b380398.txt">Table of n, a(n) for n = 1..10000</a>
%F A380398 a(n) = Sum_{d|n, gcd(d, n/d) == 1} [d in A001597], where [] is the Iverson bracket.
%F A380398 a(n) = A091050(n) - A380399(n).
%F A380398 a(n) = 1 if and only if n is squarefree (A005117).
%F A380398 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 - Sum_{k>=2} mu(k)*(zeta(k)/zeta(k+1) - 1) = 1.49341326536904597349..., where mu is the Moebius function (A008683).
%e A380398 a(4) = 2 since 4 have 2 unitary divisors that are perfect powers, 1 and 4 = 2^2.
%e A380398 a(72) = 3 since 72 have 3 unitary divisors that are perfect powers, 1, 8 = 2^3, and 9 = 3^2.
%t A380398 ppQ[n_] := n == 1 || GCD @@ FactorInteger[n][[;; , 2]] > 1; a[n_] := DivisorSum[n, 1 &, CoprimeQ[#, n/#] && ppQ[#] &]; Array[a, 100]
%o A380398 (PARI) a(n) = sumdiv(n, d, gcd(d, n/d) == 1 && (d == 1 || ispower(d)));
%Y A380398 Cf. A001597, A005117, A008683, A061742, A077610, A091050, A118914, A323308, A380399, A380400.
%Y A380398 Cf. A007424, A278908, A307848, A323308, A358260, A365549, A368978, A369163.
%K A380398 nonn,easy
%O A380398 1,4
%A A380398 _Amiram Eldar_, Jan 23 2025