A365211 The sum of divisors d of n such that gcd(d, n/d) is a 5-rough number (A007310).

1, 3, 4, 5, 6, 12, 8, 9, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 36, 31, 42, 28, 40, 30, 72, 32, 33, 48, 54, 48, 50, 38, 60, 56, 54, 42, 96, 44, 60, 60, 72, 48, 68, 57, 93, 72, 70, 54, 84, 72, 72, 80, 90, 60, 120, 62, 96, 80, 65, 84, 144, 68

Offset

1,2

Comments

First differs from A034448 at n = 25.

The number of these divisors is A365210(n).

Links

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

Formula

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

a(n) <= A000203(n), with equality if and only if n is neither divisible by 4 nor by 9.

a(n) >= A034448(n), with equality if and only if n is not divisible by a square of a prime >= 5.

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

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

Sum_{k=1..n} a(k) ~ c * n^2, where c = 91*Pi^2/1296 = 0.69300463... .

Mathematica

f[p_, e_] := If[p <= 3 , 1 + p^e, (p^(e+1)-1)/(p-1)]; 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, 1 + f[i, 1]^f[i, 2], (f[i, 1]^(f[i, 2]+1)-1)/(f[i, 1]-1))); }

Crossrefs

Cf. A000203, A007310, A034448, A065330, A065331, A069184, A186099, A365210.

Sequence in context: A378433 A378434 A034448 * A365172 A331107 A069184

Adjacent sequences: A365208 A365209 A365210 * A365212 A365213 A365214

Keyword

nonn,easy,mult

Author

Amiram Eldar, Aug 26 2023

Status

approved