A390665 - OEIS

%I #9 Nov 14 2025 09:39:56

%S 1,9,28,9,126,252,344,9,28,1134,1332,252,2198,3096,3528,65,4914,252,

%T 6860,1134,9632,11988,12168,252,126,19782,28,3096,24390,31752,29792,

%U 65,37296,44226,43344,252,50654,61740,61544,1134,68922,86688,79508,11988,3528,109512

%N The sum of the unitary divisors of the smallest cube divisible by n.

%C First differs from A351266, A369721 and A369759 at n = 16.

%H Amiram Eldar, <a href="/A390665/b390665.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A034448(A053149(n)).

%F a(n) >= A034448(n) with equality if and only if n is a cube (A000578).

%F Multiplicative with a(p^e) = p^(e + ((3-e) mod 3)) + 1.

%F Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(4) * zeta(9) * Product_{p prime} (1 - 1/p^2 - 1/p^9 + 1/p^10 - 1/p^13 + 1/p^14) = 0.658357292864217429508... .

%t f[p_, e_] := p^(e + Mod[3 - e, 3]) + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(f[i,2] + (3-f[i,2])%3) + 1);}

%Y Cf. A000578, A034448, A053149, A351266, A365479, A369718, A369721, A369759, A390663, A390666.

%Y Cf. A013662, A013667.

%K nonn,mult,easy

%O 1,2

%A _Amiram Eldar_, Nov 14 2025