A336651 - OEIS

%I #20 Sep 14 2023 02:31:32

%S 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,

%T 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,

%U 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

%N Odd part of n divided by its largest squarefree divisor.

%C 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.

%H Antti Karttunen, <a href="/A336651/b336651.txt">Table of n, a(n) for n = 1..65537</a>

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

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

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

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

%F 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

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

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

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

%K nonn,easy,mult

%O 1,9

%A _Antti Karttunen_, Jul 30 2020