A365346 - OEIS

%I #10 Sep 02 2023 16:59:37

%S 1,7,13,7,31,91,57,31,13,217,133,91,183,399,403,31,307,91,381,217,741,

%T 931,553,403,31,1281,121,399,871,2821,993,127,1729,2149,1767,91,1407,

%U 2667,2379,961,1723,5187,1893,931,403,3871,2257,403,57,217,3991,1281,2863

%N The sum of divisors of the smallest square divisible by n.

%C The number of these divisors is A365345(n).

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

%H Amiram Eldar, <a href="/A365346/b365346.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%F 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)).

%F 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... .

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

%o (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));}

%o (PARI) a(n) = sigma(n*core(n)); \\ _Michel Marcus_, Sep 02 2023

%Y Cf. A000203, A053143, A365345, A365347.

%K nonn,easy,mult

%O 1,2

%A _Amiram Eldar_, Sep 02 2023