A336651 Odd part of n divided by its largest squarefree divisor.

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 5, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 5, 1, 1, 1, 1, 1, 27, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 7, 3, 5, 1, 1, 1, 1, 1

Offset: 1

Author

Antti Karttunen, Jul 30 2020

Keywords

• nonn
• easy
• mult

Comments

The name can be parsed either as "Odd part of {n divided by its largest squarefree divisor}" or "Odd part of n, divided by its largest squarefree divisor". Because A000265 and A003557 commute, both interpretations yield equal results.

Links

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

Crossrefs

Cf. A000203, A000265, A003557, A007947, A336649, A336652.

Programs

Mathematica

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

PARI

A336651(n) = { my(f=factor(n)); prod(i=1, #f~, if(2==f[i,1],1,f[i,1]^(f[i,2]-1))); };

Formula

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

a(n) = A003557(A000265(n)) = A000265(A003557(n)).

a(n) = A000265(n) / A204455(n).

A000203(a(n)) = A336649(n).

Dirichlet g.f.: (1 - 1/(2^s-1)^2) * zeta(s-1) * Product_{p prime} (1 - 1/p^(s-1) + 1/p^s). - Amiram Eldar, Sep 14 2023