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A384556 The sum of the exponential divisors of n that are cubefree.

%I A384556 #12 Jun 03 2025 15:17:01
%S A384556 1,2,3,6,5,6,7,2,12,10,11,18,13,14,15,6,17,24,19,30,21,22,23,6,30,26,
%T A384556 3,42,29,30,31,2,33,34,35,72,37,38,39,10,41,42,43,66,60,46,47,18,56,
%U A384556 60,51,78,53,6,55,14,57,58,59,90,61,62,84,6,65,66,67,102,69
%N A384556 The sum of the exponential divisors of n that are cubefree.
%C A384556 The number of these divisors is A056624(n), and the largest of them is A066990(n).
%H A384556 Amiram Eldar, <a href="/A384556/b384556.txt">Table of n, a(n) for n = 1..10000</a>
%F A384556 Multiplicative with a(p^e) = p if e is odd, and p+p^2 is e is even.
%F A384556 a(n) <= A051377(n), with equality if and only if n is cubefree (A004709).
%F A384556 Dirichlet g.f.: zeta(2*s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-2) + 1/p^(2*s-1) - 1/p^(2*s)).
%F A384556 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4) * Product_{p prime} (1 - 2/p^4 + 1/p^5) = 0.95692470821076622881...
%t A384556 f[p_, e_] := If[OddQ[e], p, p + p^2]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A384556 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1] + if(!(f[i,2] % 2), f[i,1]^2));}
%o A384556 (Python)
%o A384556 from math import prod
%o A384556 from sympy import factorint
%o A384556 def A384556(n): return prod(p*(1+p*(e&1^1)) for p, e in factorint(n).items()) # _Chai Wah Wu_, Jun 03 2025
%Y A384556 Cf. A004709, A013662, A051377, A056624, A066990, A073185, A322791, A384554.
%K A384556 nonn,easy,mult
%O A384556 1,2
%A A384556 _Amiram Eldar_, Jun 03 2025