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A360721 a(n) is the number of infinitary divisors of n that are powerful (A001694).

%I A360721 #11 May 21 2025 03:21:16
%S A360721 1,1,1,2,1,1,1,3,2,1,1,2,1,1,1,2,1,2,1,2,1,1,1,3,2,1,3,2,1,1,1,3,1,1,
%T A360721 1,4,1,1,1,3,1,1,1,2,2,1,1,2,2,2,1,2,1,3,1,3,1,1,1,2,1,1,2,4,1,1,1,2,
%U A360721 1,1,1,6,1,1,2,2,1,1,1,2,2,1,1,2,1,1,1
%N A360721 a(n) is the number of infinitary divisors of n that are powerful (A001694).
%H A360721 Amiram Eldar, <a href="/A360721/b360721.txt">Table of n, a(n) for n = 1..10000</a>
%H A360721 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%H A360721 <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>.
%F A360721 Multiplicative with a(p^e) = 2^A000120(e) - (e mod 2).
%F A360721 a(n) <= A037445(n) with equality if and only if n is a square.
%F A360721 a(n) <= A005361(n) with equality if and only if n is not in A360723.
%F A360721 Sum_{k=1..n} a(k) ~ c * n, where c = Product_{p prime} ((1-1/p) * Sum_{k>=1} ((2^A000120(k)- k mod 2)/p^k)) = 1.72717... .
%t A360721 f[p_, e_] := 2^DigitCount[e, 2, 1] - Mod[e, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A360721 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, 2^hammingweight(f[i, 2]) - f[i, 2]%2);}
%Y A360721 Cf. A000120, A001694, A037445, A077609, A360723.
%Y A360721 Similar sequences: A005361 (number of powerful divisors), A323308 (number of unitary powerful divisors).
%K A360721 nonn,mult,easy
%O A360721 1,4
%A A360721 _Amiram Eldar_, Feb 18 2023