A183105 a(n) = product of divisors of n that are not perfect powers.

1, 2, 3, 2, 5, 36, 7, 2, 3, 100, 11, 432, 13, 196, 225, 2, 17, 648, 19, 2000, 441, 484, 23, 10368, 5, 676, 3, 5488, 29, 810000, 31, 2, 1089, 1156, 1225, 7776, 37, 1444, 1521, 80000, 41, 3111696, 43, 21296, 10125, 2116, 47, 497664, 7, 5000, 2601, 35152, 53, 34992, 3025, 307328, 3249, 3364, 59, 11664000000, 61, 3844, 27783, 2, 4225, 18974736, 67, 78608, 4761, 24010000, 71, 13436928

Offset: 1

Author

Jaroslav Krizek, Dec 25 2010

Keywords

• nonn
• look

Comments

Sequence is not the same as A183103: a(72) = 13436928, A183103(72) = 186624.

Not multiplicative since a(2)*a(3) <> a(6). - R. J. Mathar, Jun 07 2011

Examples

			For n = 12, set of such divisors is {2, 3, 6, 12}; a(12) = 1*2*3*6*12 = 432.
		

Links

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

Crossrefs

Cf. A001597, A007955, A183101, A183103, A183104.

Programs

Maple

isA001597 := proc(n) local e ; e := seq(op(2,p),p=ifactors(n)[2]) ; return ( igcd(e) >=2 ) ; end proc:
A183105 := proc(n) local a,d; a := 1 ; for d in numtheory[divisors](n) do if not isA001597(d) then a := a*d; end if; end do; a ; end proc:
seq(A183105(n),n=1..72) ; # R. J. Mathar, Jun 07 2011

Mathematica

perfectPowerQ[n_] := GCD @@ FactorInteger[n][[All, 2]] > 1;
a[n_] := Times @@ Select[Divisors[n], !perfectPowerQ[#]&];
Table[a[n], {n, 1, 72}] (* Jean-François Alcover, May 31 2024 *)

PARI

A183105(n) = { my(m=1); fordiv(n, d, if(!ispower(d), m *= d)); m; }; \\ Antti Karttunen, Oct 07 2017

Formula

a(n) = A007955(n) / A183104(n).

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