A336649 Sum of divisors of A336651(n) (odd part of n divided by its largest squarefree divisor).

1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 6, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 8, 6, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 6, 1, 1, 1, 1, 1, 40, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 8, 4, 6, 1, 1, 1, 1, 1

Offset: 1

Author

Antti Karttunen, Jul 30 2020

Keywords

• nonn
• mult

Links

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

Crossrefs

Cf. A000203, A000265, A204455, A335341, A336651, A336652.

Programs

Mathematica

f[2, e_] := 1; f[p_, e_] := (p^e - 1)/(p-1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 07 2020 *)

PARI

A336649(n) = { my(f=factor(n)); prod(i=1, #f~, if((2==f[i,1])||(1==f[i,2]),1,(((f[i,1]^(f[i,2]))-1) / (f[i,1]-1)))); };

PARI

A000265(n) = (n>>valuation(n,2));
A335341(n) = if(1==n,n,sigma(n/factorback(factorint(n)[, 1])));
A336649(n) = A335341(A000265(n));

Formula

Multiplicative with a(2^e) = 1, a(p^1) = 1 and a(p^e) = (p^e - 1)/(p-1) if e > 1.

a(n) = A000203(A336651(n)) = A335341(A000265(n)).

a(n) = A336652(n) / A204455(n).

Dirichlet g.f.: zeta(s-1) * zeta(s) * (1 - 1/(1-2^s+2^(2*s-1))) * Product_{p prime} (1 - 1/p^(s-1) + 1/p^(2*s-1)). - Amiram Eldar, Dec 18 2023