A392170 The Dedekind psi function value of the smallest powerful number that is a multiple of n.

1, 6, 12, 6, 30, 72, 56, 12, 12, 180, 132, 72, 182, 336, 360, 24, 306, 72, 380, 180, 672, 792, 552, 144, 30, 1092, 36, 336, 870, 2160, 992, 48, 1584, 1836, 1680, 72, 1406, 2280, 2184, 360, 1722, 4032, 1892, 792, 360, 3312, 2256, 288, 56, 180, 3672, 1092, 2862, 216

Offset

1,2

Links

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

Formula

a(n) = A001615(A197863(n)).

Multiplicative with a(p^e) = (p+1) * p^(max(e, 2)-1).

Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-3) + 1/p^(2*s-1)).

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Product_{p prime} (1 - 1/p^3 + 1/p^4) = 0.90470892696874750603... .

Sum_{n>=1} 1/a(n) = zeta(2) * Product_{p prime} (1 + 1/p^2 - 1/p^3) = 2.14823170440117252186... .

Mathematica

f[p_, e_] := (p+1) * p^(Max[e, 2]-1); a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100]

Prog

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

Crossrefs

Cf. A001615, A013661, A197863, A390755, A392089.

Sequence in context: A050496 A262617 A303226 * A392168 A360877 A295122

Adjacent sequences: A392167 A392168 A392169 * A392171 A392172 A392173

Keyword

nonn,mult,easy

Author

Amiram Eldar, Jan 02 2026

Status

approved