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

Author

Amiram Eldar, Sep 02 2023

Keywords

• nonn
• easy
• mult

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

Crossrefs

Cf. A000203, A019554, A322483, A365346.

Cf. A002117, A065465.

Programs

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]

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

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... .