A183104 a(n) = product of divisors of n that are perfect powers.

1, 1, 1, 4, 1, 1, 1, 32, 9, 1, 1, 4, 1, 1, 1, 512, 1, 9, 1, 4, 1, 1, 1, 32, 25, 1, 243, 4, 1, 1, 1, 16384, 1, 1, 1, 1296, 1, 1, 1, 32, 1, 1, 1, 4, 9, 1, 1, 512, 49, 25, 1, 4, 1, 243, 1, 32, 1, 1, 1, 4, 1, 1, 9, 1048576, 1, 1, 1, 4, 1, 1, 1, 10368

Offset: 1

Author

Jaroslav Krizek, Dec 25 2010

Keywords

• nonn

Comments

Sequence is not the same as A183102: a(72) = 10368, A183102(72) = 746496.

Not multiplicative, as a(4)*a(9) <> a(36). - R. J. Mathar, Jun 07 2011

Examples

			For n = 12, set of such divisors is {1, 4}; a(12) = 1*4 = 4.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16385

Crossrefs

Cf. A001597, A007955, A091051, A183102, A183105.

Programs

Maple

isA001597 := proc(n) local e ; e := seq(op(2,p),p=ifactors(n)[2]) ; return ( igcd(e) >=2 ) ; end proc:
A183104 := proc(n) local a,d; a := 1 ; for d in numtheory[divisors](n) do if isA001597(d) then a := a*d; end if; end do; a ; end proc:
seq(A183104(n),n=1..72) ; # R. J. Mathar, Jun 07 2011

Mathematica

perfPQ[n_]:=GCD@@FactorInteger[n][[All,2]]>1; Table[Times@@Select[Divisors[n],perfPQ[#]&],{n,120}] (* Harvey P. Dale, Mar 07 2024 *)

PARI

A183104(n) = { my(m=1); fordiv(n, d, if(ispower(d), m *= d)); m; }; \\ Antti Karttunen, Oct 07 2017

Formula

a(n) = A007955(n) / A183105(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.