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A360720 a(n) is the sum of unitary divisors of n that are powerful (A001694).

%I A360720 #34 Jun 19 2025 03:28:24
%S A360720 1,1,1,5,1,1,1,9,10,1,1,5,1,1,1,17,1,10,1,5,1,1,1,9,26,1,28,5,1,1,1,
%T A360720 33,1,1,1,50,1,1,1,9,1,1,1,5,10,1,1,17,50,26,1,5,1,28,1,9,1,1,1,5,1,1,
%U A360720 10,65,1,1,1,5,1,1,1,90,1,1,26,5,1,1,1,17,82
%N A360720 a(n) is the sum of unitary divisors of n that are powerful (A001694).
%C A360720 The number of these divisors is given by A323308.
%H A360720 Amiram Eldar, <a href="/A360720/b360720.txt">Table of n, a(n) for n = 1..10000</a>
%H A360720 Vaclav Kotesovec, <a href="/A360720/a360720.jpg">Plot of Sum_{k=1..n} a(k) / n^(3/2) for n = 1..10^8</a>.
%H A360720 <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>.
%F A360720 Multiplicative with a(p) = 1 and a(p^e) = p^e + 1 for e > 1.
%F A360720 a(n) <= A034448(n), with equality if and only if n is powerful (A001694).
%F A360720 a(n) <= A183097(n), with equality if and only if n is cubefree (A004709).
%F A360720 Dirichlet g.f.: zeta(s)*zeta(s-1)*Product_{p prime} (1 - p^(1-s) + p^(2-2*s) - p^(2-3*s)).
%F A360720 From _Vaclav Kotesovec_, Feb 18 2023: (Start)
%F A360720 Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{primes p} (1 - p^(3-4*s) - p^(2-3*s) + p^(3-3*s)).
%F A360720 Sum_{k=1..n} a(k) ~ c * zeta(3/2) * n^(3/2) / 3, where c = Product_{primes p} (1 + 1/p^(3/2) - 1/p^(5/2) - 1/p^3) = 1.48039182258752809541724060173644... (End)
%F A360720 a(n) = A034448(A057521(n)) (the sum of unitary divisors of the powerful part of n). - _Amiram Eldar_, Dec 12 2023
%F A360720 a(n) = A034448(n)/A092261(n). - _Amiram Eldar_, Jun 19 2025
%t A360720 f[p_, e_] := If[e == 1, 1, p^e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A360720 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, 1, f[i, 1]^f[i, 2] + 1));}
%o A360720 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - p^3*X^4 - p^2*X^3 + p^3*X^3) / ((1 - X) * (1 - p^2*X^2)))[n], ", ")) \\ _Vaclav Kotesovec_, Feb 18 2023
%Y A360720 Cf. A001694, A004709, A034444, A034448, A057521, A077610, A092261, A323308.
%Y A360720 Similar sequences: A183097, A360722.
%K A360720 nonn,mult,easy
%O A360720 1,4
%A A360720 _Amiram Eldar_, Feb 18 2023