A365347 The sum of divisors of the smallest number whose square is divisible by n.

1, 3, 4, 3, 6, 12, 8, 7, 4, 18, 12, 12, 14, 24, 24, 7, 18, 12, 20, 18, 32, 36, 24, 28, 6, 42, 13, 24, 30, 72, 32, 15, 48, 54, 48, 12, 38, 60, 56, 42, 42, 96, 44, 36, 24, 72, 48, 28, 8, 18, 72, 42, 54, 39, 72, 56, 80, 90, 60, 72, 62, 96, 32, 15, 84, 144, 68, 54

Offset

1,2

Comments

The number of divisors of the smallest number whose square is divisible by n is A322483(n).

The sum of divisors of the smallest square divisible by n is A365346(n).

Links

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

Formula

a(n) = A000203(A019554(n)).

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

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * zeta(3) * Product_{p prime} (1 - 1/(p^2*(p+1))) = (1/2) * A002117 * A065465 = 0.529814898136... .

Mathematica

f[p_, e_] := (p^((e + Mod[e, 2])/2 + 1) - 1)/(p - 1); 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]^((f[i, 2] + f[i, 2]%2)/2 + 1) - 1)/(f[i, 1] - 1)); }
(PARI) a(n) = sigma(n/core(n, 1)[2]); \\ Michel Marcus, Sep 02 2023

Crossrefs

Cf. A000203, A019554, A322483, A365346.

Cf. A002117, A065465.

Sequence in context: A369889 A365481 A000113 * A069915 A033634 A366764

Adjacent sequences: A365344 A365345 A365346 * A365348 A365349 A365350

Keyword

nonn,easy,mult

Author

Amiram Eldar, Sep 02 2023

Status

approved