A385136 The sum of divisors d of n such that n/d is a cubefull number (A036966).

1, 2, 3, 4, 5, 6, 7, 9, 9, 10, 11, 12, 13, 14, 15, 19, 17, 18, 19, 20, 21, 22, 23, 27, 25, 26, 28, 28, 29, 30, 31, 39, 33, 34, 35, 36, 37, 38, 39, 45, 41, 42, 43, 44, 45, 46, 47, 57, 49, 50, 51, 52, 53, 56, 55, 63, 57, 58, 59, 60, 61, 62, 63, 79, 65, 66, 67, 68

Offset

1,2

Links

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

Formula

Multiplicative with a(p) = p and a(p^e) = (p^(e+1) - p^e + p^(e-2) - 1)/(p-1) for e >= 2.

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

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2) * Product_{p prime} (1 - 1/p^2 + 1/p^6) = 1.022486596136980366... .

Mathematica

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

Prog

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

Crossrefs

Cf. A013661, A036966, A385005.

The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), A385134 (biquadratefree), A385135 (exponentially odd), this sequence (cubefull), A385137 (3-smooth), A385138 (5-rough), A385139 (exponentially 2^n).

Sequence in context: A063932 A323785 A327626 * A179464 A071191 A300903

Adjacent sequences: A385133 A385134 A385135 * A385137 A385138 A385139

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jun 19 2025

Status

approved