A392169 The Dedekind psi function value of the smallest cube divisible by n.

1, 12, 36, 12, 150, 432, 392, 12, 36, 1800, 1452, 432, 2366, 4704, 5400, 96, 5202, 432, 7220, 1800, 14112, 17424, 12696, 432, 150, 28392, 36, 4704, 25230, 64800, 30752, 96, 52272, 62424, 58800, 432, 52022, 86640, 85176, 1800, 70602, 169344, 81356, 17424, 5400, 152352

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

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

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

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... .

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... .

Mathematica

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

Prog

(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)); }

Crossrefs

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

Sequence in context: A007794 A008653 A392171 * A038006 A205967 A203378

Adjacent sequences: A392166 A392167 A392168 * A392170 A392171 A392172

Keyword

nonn,mult,easy

Author

Amiram Eldar, Jan 02 2026

Status

approved