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

Author

Amiram Eldar, Jun 19 2025

Keywords

• nonn
• easy
• mult

Links

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

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).

Programs

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]

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)));}

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... .