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

1,4

Links

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

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.

Example

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

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 *)

Prog

(Haskell)
a178649 n = div (a007955 n) (a078599 n)
-- Reinhard Zumkeller, Feb 06 2012
(PARI) a(n) = my(p=1); fordiv(n, d, if (!issquarefree(d), p*=d)); p; \\ Michel Marcus, Jul 06 2022

Crossrefs

Cf. A007947, A007955, A078599.

Sequence in context: A112622 A183104 A183102 * A353976 A363919 A368000

Adjacent sequences: A178646 A178647 A178648 * A178650 A178651 A178652

Keyword

nonn

Author

Jaroslav Krizek, Dec 25 2010

Status

approved