login
A344875
Multiplicative with a(2^e) = 2^(1+e) - 1, and a(p^e) = p^e - 1 for odd primes p.
19
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
1,2
LINKS
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
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 *)
PROG
(PARI) A344875(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^(f[2, i]+(2==f[1, i]))-1)); };
(Python 3.8+)
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
KEYWORD
nonn,mult
AUTHOR
Antti Karttunen, Jun 03 2021
STATUS
approved