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A384555 The largest infinitary divisor of n that is cubefree.

%I A384555 #7 Jun 03 2025 11:57:35
%S A384555 1,2,3,4,5,6,7,4,9,10,11,12,13,14,15,1,17,18,19,20,21,22,23,12,25,26,
%T A384555 9,28,29,30,31,2,33,34,35,36,37,38,39,20,41,42,43,44,45,46,47,3,49,50,
%U A384555 51,52,53,18,55,28,57,58,59,60,61,62,63,4,65,66,67,68,69
%N A384555 The largest infinitary divisor of n that is cubefree.
%C A384555 The number of these divisors is A368883(n), and their sum is A384554(n).
%H A384555 Amiram Eldar, <a href="/A384555/b384555.txt">Table of n, a(n) for n = 1..10000</a>
%F A384555 Multiplicative with a(p^e) = 1 if e == 0 (mod 4), p if e == 1 (mod 4), p^2 if e == 2 or 3 (mod 4).
%F A384555 a(n) = n if and only if n is cubefree (A004709).
%F A384555 Dirichlet g.f.: zeta(4*s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-2) + 1/p^(3*s-2)).
%F A384555 Sum_{k=1..n} a(k) = c * n^2 / 2, where c = zeta(8) * Product_{p prime} (1 - 1/p^3 + 1/p^4 - 1/p^5) = 0.87406992849637563411... .
%t A384555 f[p_, e_] := Switch[Mod[e, 4], 0, 1, 1, p, 2, p^2, 3, p^2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A384555 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; [1, p, p^2, p^2][e%4+1]);}
%Y A384555 Cf. A004709, A007948, A013666, A077609, A368883, A384554.
%K A384555 nonn,easy,mult
%O A384555 1,2
%A A384555 _Amiram Eldar_, Jun 03 2025