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

1,2

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

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... .

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]

Prog

(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])); }

Crossrefs

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

Cf. A002117, A013661, A306633.

Sequence in context: A325317 A325316 A227131 * A097863 A287926 A097012

Adjacent sequences: A365206 A365207 A365208 * A365210 A365211 A365212

Keyword

nonn,easy,mult

Author

Amiram Eldar, Aug 26 2023

Status

approved