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

1, 7, 13, 7, 31, 91, 57, 31, 13, 217, 133, 91, 183, 399, 403, 31, 307, 91, 381, 217, 741, 931, 553, 403, 31, 1281, 121, 399, 871, 2821, 993, 127, 1729, 2149, 1767, 91, 1407, 2667, 2379, 961, 1723, 5187, 1893, 931, 403, 3871, 2257, 403, 57, 217, 3991, 1281, 2863

Offset

1,2

Comments

The number of these divisors is A365345(n).

The sum of divisors of the square root of the smallest square divisible by n is A365347(n).

Links

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

Formula

a(n) = A000203(A053143(n)).

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

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

Sum_{k=1..n} a(k) ~ c * n^3, where c = (Pi^2/45) * zeta(3) * Product_{p prime} (1 + 1/p^2 - 1/p^3) = 0.344306233314... .

Mathematica

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

Crossrefs

Cf. A000203, A053143, A365345, A365347.

Sequence in context: A158622 A367866 A369717 * A215990 A122874 A066003

Adjacent sequences: A365343 A365344 A365345 * A365347 A365348 A365349

Keyword

nonn,easy,mult

Author

Amiram Eldar, Sep 02 2023

Status

approved