A344879 a(n) = A344875(n) / A344878(n), where A344875(n) is multiplicative with a(2^e) = 2^(1+e) - 1, and a(p^e) = p^e -1 for odd primes p, and A344878(n) gives the least common multiple of the same factors.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 3, 2, 1, 1, 6, 1, 1, 4, 1, 1, 1, 1, 3, 2, 1, 1, 1, 2, 3, 2, 1, 1, 2, 1, 3, 2, 1, 4, 2, 1, 1, 2, 6, 1, 1, 1, 3, 2, 1, 2, 6, 1, 1, 1, 1, 1, 2, 4, 3, 2, 5, 1, 4, 6, 1, 2, 1, 2, 1, 1, 3, 2, 1, 1, 2, 1, 3, 4

Offset: 1

Author

Antti Karttunen, Jun 03 2021

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A344875, A344878.

Programs

Mathematica

f[2, e_] := 2^(e + 1) - 1; f[p_, e_] := p^e - 1; a[1] = 1; a[n_] := Times @@ (fct = f @@@ FactorInteger[n])/LCM @@ fct; 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)); };
A344878(n) = if(1==n,n, my(f=factor(n)~); lcm(vector(#f, i, (f[1, i]^(f[2, i]+(2==f[1, i]))-1))));
A344879(n) = (A344875(n) / A344878(n));

Python

from math import prod, lcm
from sympy import factorint
def A344879(n): return prod(a := tuple(p**(e+int(p==2))-1 for p, e in factorint(n).items()))//lcm(*a) # Chai Wah Wu, Jun 15 2022