A392012 The sum of the exponential divisors of the cubefree numbers.

1, 2, 3, 6, 5, 6, 7, 12, 10, 11, 18, 13, 14, 15, 17, 24, 19, 30, 21, 22, 23, 30, 26, 42, 29, 30, 31, 33, 34, 35, 72, 37, 38, 39, 41, 42, 43, 66, 60, 46, 47, 56, 60, 51, 78, 53, 55, 57, 58, 59, 90, 61, 62, 84, 65, 66, 67, 102, 69, 70, 71, 73, 74, 90, 114, 77, 78

Offset

1,2

Comments

Also, the sum of the exponential unitary divisors of the cubefree numbers.

The number of these divisors is A382419(n).

Links

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

Formula

a(n) = A051377(A004709(n)).

a(n) = A057723(A004709(n)).

a(n) = A322857(A004709(n)).

Sum_{k=1..n} a(k) ~ c * n^2, where c = zeta(3)/(2*zeta(4)) = 0.5553132... .

Mathematica

f[p_, e_] := If[e < 3, Sum[p^k, {k, 1, e}], 0]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 100] , # > 0 &]

Prog

(PARI) s(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] < 3, sum(k = 1, f[i, 2], f[i, 1]^k), 0)); }
list(lim) = select(x -> x > 0, vector(lim, i, s(i)));

Crossrefs

Cf. A002117, A004709, A013662, A051377, A057723, A322791, A322857, A382419.

Sequence in context: A322857 A361175 A051377 * A392016 A369319 A336465

Adjacent sequences: A392009 A392010 A392011 * A392013 A392014 A392015

Keyword

nonn,easy

Author

Amiram Eldar, Dec 27 2025

Status

approved