A366148 The sum of divisors of the cubefree part of n (A360539).

1, 3, 4, 7, 6, 12, 8, 1, 13, 18, 12, 28, 14, 24, 24, 1, 18, 39, 20, 42, 32, 36, 24, 4, 31, 42, 1, 56, 30, 72, 32, 1, 48, 54, 48, 91, 38, 60, 56, 6, 42, 96, 44, 84, 78, 72, 48, 4, 57, 93, 72, 98, 54, 3, 72, 8, 80, 90, 60, 168, 62, 96, 104, 1, 84, 144, 68, 126, 96

Offset: 1

Author

Amiram Eldar, Oct 01 2023

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Cf. A000203, A004709, A036966, A092261, A360539, A366077, A366078, A366146, A366147.

Programs

Mathematica

f[p_, e_] := If[e < 3, (p^(e+1)-1)/(p-1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] < 3, (f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1), 1));}

Formula

a(n) = A000203(A360539(n)).

a(n) = A000203(n)/A366146(n).

Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if e <= 2, and 1 otherwise.

a(n) >= 1, with equality if and only if n is cubefull (A036966).

a(n) <= A000203(n), with equality if and only if n is cubefree (A004709).

Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-2) - 1/p^(3*s-2) - 1/p^(3*s-1)). CHECK

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (p/(p+1) + 1/p - 1/p^4) = 0.63884633697952950095... .