A392169 - OEIS

%I #8 Jan 03 2026 04:38:30

%S 1,12,36,12,150,432,392,12,36,1800,1452,432,2366,4704,5400,96,5202,

%T 432,7220,1800,14112,17424,12696,432,150,28392,36,4704,25230,64800,

%U 30752,96,52272,62424,58800,432,52022,86640,85176,1800,70602,169344,81356,17424,5400,152352

%N The Dedekind psi function value of the smallest cube divisible by n.

%H Amiram Eldar, <a href="/A392169/b392169.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A001615(A053149(n)).

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

%F Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(9) * Product_{p prime} (1 - 1/p^3 + 1/p^5 - 1/p^7 + 1/p^10 - 1/p^11) = 0.85995977863257911562... .

%F Sum_{n>=1} 1/a(n) = zeta(2) * zeta(3) * Product_{p prime} (1 - 1/p^2 + 2/p^3 - 3/p^4 + 1/p^5) = 1.44337397853300642934... .

%t f[p_, e_] := (p+1) * p^(e - 1 + Mod[3-e, 3]); 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]+1) * f[i,1]^(f[i,2] - 1 + (3-f[i,2])%3));}

%Y Cf. A001615, A002117, A013661, A013667, A053149, A390754, A392088.

%K nonn,mult,easy

%O 1,2

%A _Amiram Eldar_, Jan 02 2026