A385137 The sum of divisors d of n such that n/d is a 3-smooth number (A003586).

1, 3, 4, 7, 5, 12, 7, 15, 13, 15, 11, 28, 13, 21, 20, 31, 17, 39, 19, 35, 28, 33, 23, 60, 25, 39, 40, 49, 29, 60, 31, 63, 44, 51, 35, 91, 37, 57, 52, 75, 41, 84, 43, 77, 65, 69, 47, 124, 49, 75, 68, 91, 53, 120, 55, 105, 76, 87, 59, 140, 61, 93, 91, 127, 65, 132

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. A003586, A064987, A072044, A072045, A072079.

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), A385136 (cubefull), this sequence (3-smooth), A385138 (5-rough), A385139 (exponentially 2^n).

Programs

Mathematica

f[p_, e_] := If[p < 5, (p^(e+1) - 1)/(p - 1), p^e]; 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(p < 5, (p^(e + 1) - 1)/(p - 1), p^e));}

Formula

a(n) = A064987(n)/A385138(n).

Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if p <= 3, and p^e if p >= 5.

In general, the sum of divisors d of n such that n/d is q-smooth (not divisible by a prime larger than q) is multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if p <= q, and p^e if p > q.

Dirichlet g.f.: zeta(s-1) / ((1 - 1/2^s) * (1 - 1/3^s)).

In general, the sum of divisors d of n such that n/d is q-smooth has Dirichlet g.f.: zeta(s-1) / Product_{p prime <= q} (1 - 1/q^s).

Sum_{k=1..n} a(k) ~ (3/4)*n^2.

In general, the sum of divisors d of n such that n/d is prime(k)-smooth has an average order c * n^2 / 2, where c = A072044(k-1)/A072045(k-1) for k >= 2.