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A369721 The sum of unitary divisors of the smallest cubefull number that is a multiple of n.

%I A369721 #7 Jan 30 2024 04:42:23
%S A369721 1,9,28,9,126,252,344,9,28,1134,1332,252,2198,3096,3528,17,4914,252,
%T A369721 6860,1134,9632,11988,12168,252,126,19782,28,3096,24390,31752,29792,
%U A369721 33,37296,44226,43344,252,50654,61740,61544,1134,68922,86688,79508,11988,3528,109512
%N A369721 The sum of unitary divisors of the smallest cubefull number that is a multiple of n.
%H A369721 Amiram Eldar, <a href="/A369721/b369721.txt">Table of n, a(n) for n = 1..10000</a>
%F A369721 a(n) = A034448(A356193(n)).
%F A369721 Multiplicative with a(p) = p^3 + 1 for e <= 2, and a(p^e) = p^e + 1 for e >= 3.
%F A369721 a(n) >= A034448(n), with equality if and only if n is cubefull (A036966).
%F A369721 Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 + 1/p^(s-3) - 1/p^(s-1) - 1/p^(2*s-4)  + 1/p^(4*s-4) - 1/p^(4*s-3) ).
%F A369721 Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(3) * zeta(4) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^5 + 1/p^12 - 2/p^13 + 1/p^14) = 0.65803546696642353777... .
%t A369721 f[p_, e_] := If[e <= 2, p^3 + 1, p^e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]
%o A369721 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] <= 2, 1 + f[i,1]^3, 1 + f[i,1]^f[i,2]));}
%Y A369721 Cf. A034448, A036966, A356193, A369718, A369719, A369720.
%Y A369721 Cf. A002117, A013662, A183700.
%K A369721 nonn,easy,mult
%O A369721 1,2
%A A369721 _Amiram Eldar_, Jan 30 2024