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

1,2

Links

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

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.

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]

Prog

(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)); }

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).

Sequence in context: A338285 A050197 A003975 * A285896 A082226 A010613

Adjacent sequences: A385134 A385135 A385136 * A385138 A385139 A385140

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jun 19 2025

Status

approved