A178649 a(n) = product of nonsquarefree divisors of n.

1, 1, 1, 4, 1, 1, 1, 32, 9, 1, 1, 48, 1, 1, 1, 512, 1, 162, 1, 80, 1, 1, 1, 9216, 25, 1, 243, 112, 1, 1, 1, 16384, 1, 1, 1, 279936, 1, 1, 1, 25600, 1, 1, 1, 176, 405, 1, 1, 7077888, 49, 1250, 1, 208

Offset: 1

Author

Jaroslav Krizek, Dec 25 2010

Keywords

• nonn

Examples

			For n = 16, set of such divisors is {4, 8, 16}; a(16) = 4*8*16 = 512.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007947, A007955, A078599.

Programs

Haskell

a178649 n = div (a007955 n) (a078599 n)
-- Reinhard Zumkeller, Feb 06 2012

Mathematica

Table[Times@@Select[Divisors[n],!SquareFreeQ[#]&],{n,60}] (* Harvey P. Dale, Nov 04 2020 *)
a[n_] := n^(DivisorSigma[0, n]/2) / (Times @@ FactorInteger[n][[;;,1]])^(2^(PrimeNu[n]-1)); Array[a, 100] (* Amiram Eldar, Jul 06 2022 *)

PARI

a(n) = my(p=1); fordiv(n, d, if (!issquarefree(d), p*=d)); p; \\ Michel Marcus, Jul 06 2022

Formula

a(n) = A007955(n) / A078599(n) = A007955(n) / A007955(A007947(n)).

a(1) = 1, a(p) = 1, a(pq) = 1, a(pq...z) = 1, a(p^k) = p^(1/2*k*(k+1)-1), for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.