A392168 The Dedekind psi function value of the smallest square divisible by n.

1, 6, 12, 6, 30, 72, 56, 24, 12, 180, 132, 72, 182, 336, 360, 24, 306, 72, 380, 180, 672, 792, 552, 288, 30, 1092, 108, 336, 870, 2160, 992, 96, 1584, 1836, 1680, 72, 1406, 2280, 2184, 720, 1722, 4032, 1892, 792, 360, 3312, 2256, 288, 56, 180, 3672, 1092, 2862

Offset

1,2

Links

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

Formula

a(n) = A001615(A053143(n)).

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

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

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(4) * Product_{p prime} (1 - 1/p^3 + 1/p^5 - 1/p^6) = 0.91937504186949557385... .

Sum_{n>=1} 1/a(n) = zeta(2)^2 * Product_{p prime} (1 - 2/p^3 + 1/p^4) = 2.01500309855955257604... .

Mathematica

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

Crossrefs

Cf. A001615, A013661, A013662, A053143, A327171, A392087.

Sequence in context: A262617 A303226 A392170 * A360877 A295122 A103698

Adjacent sequences: A392165 A392166 A392167 * A392169 A392170 A392171

Keyword

nonn,mult,easy

Author

Amiram Eldar, Jan 02 2026

Status

approved