A365209 The sum of divisors d of n such that gcd(d, n/d) is a 3-smooth number (A003586).

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 26, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 50, 78, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 144

Offset: 1

Author

Amiram Eldar, Aug 26 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A000005 at n = 25.

The number of these divisors is A365208(n).

Links

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

Crossrefs

Cf. A000203, A003586, A034448, A065330, A065331, A365208.

Cf. A002117, A013661, A306633.

Programs

Mathematica

f[p_, e_] := If[p <= 3, (p^(e+1)-1)/(p-1), 1 + p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] <= 3, (f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1), 1 + f[i,1]^f[i,2]));}

Formula

Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) for p = 2 or 3, and a(p^e) = 1 + p^e for a prime p >= 5.

a(n) <= A000203(n), with equality if and only if n is not divisible by a square of a prime >= 5.

a(n) >= A034448(n), with equality if and only if n is neither divisible by 4 nor by 9.

a(n) = A000203(A065331(n)) * A034448(A065330(n)).

Dirichlet g.f.: (4^s/(4^s-2)) * (9^s/(9^s-3)) * zeta(s)*zeta(s-1)/zeta(2*s-1).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (54/91) * zeta(2)/zeta(3) = (54/91) * A306633 = 0.812037... .