A344875 Multiplicative with a(2^e) = 2^(1+e) - 1, and a(p^e) = p^e - 1 for odd primes p.

1, 3, 2, 7, 4, 6, 6, 15, 8, 12, 10, 14, 12, 18, 8, 31, 16, 24, 18, 28, 12, 30, 22, 30, 24, 36, 26, 42, 28, 24, 30, 63, 20, 48, 24, 56, 36, 54, 24, 60, 40, 36, 42, 70, 32, 66, 46, 62, 48, 72, 32, 84, 52, 78, 40, 90, 36, 84, 58, 56, 60, 90, 48, 127, 48, 60, 66, 112, 44, 72, 70, 120, 72, 108, 48, 126, 60, 72, 78, 124, 80, 120

Offset: 1

Author

Antti Karttunen, Jun 03 2021

Keywords

• nonn
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A047994, A065463, A153151, A344876, A344877, A344878, A344879, A344969, A345947.

Programs

Mathematica

f[2, e_] := 2^(e + 1) - 1; f[p_, e_] := p^e - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jun 03 2021 *)

PARI

A344875(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^(f[2, i]+(2==f[1, i]))-1)); };

Python

from math import prod
from sympy import factorint
def A344875(n): return prod((p**(1+e) if p == 2 else p**e)-1 for p, e in factorint(n).items()) # Chai Wah Wu, Jun 01 2022

Formula

a(n) = A344878(n) * A344879(n).

Multiplicative with a(p^e) = A153151(p^e). - Antti Karttunen, Jul 01 2021

Sum_{k=1..n} a(k) ~ c * n^2, where c = (4/5) * Product_{p prime} (1 - 1/(p*(p+1))) = (4/5) * A065463 = 0.563553... . - Amiram Eldar, Nov 18 2022