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A300330 a(n) is the product over all prime powers p^e where p^e is the highest power of p dividing n and p-1 does not divide n.

%I A300330 #17 Sep 08 2022 08:46:20
%S A300330 1,1,3,1,5,1,7,1,9,5,11,1,13,7,15,1,17,1,19,1,21,11,23,1,25,13,27,7,
%T A300330 29,5,31,1,33,17,35,1,37,19,39,1,41,1,43,11,45,23,47,1,49,25,51,13,53,
%U A300330 1,55,7,57,29,59,1,61,31,63,1,65,11,67,17,69,35,71,1
%N A300330 a(n) is the product over all prime powers p^e where p^e is the highest power of p dividing n and p-1 does not divide n.
%H A300330 Vincenzo Librandi, <a href="/A300330/b300330.txt">Table of n, a(n) for n = 1..2000</a>
%F A300330 a(n) * A193267(n) = n.
%p A300330 A300330 := proc(n) local P, F, f, divides; divides := (a,b) -> is(irem(b,a) = 0):
%p A300330 P := 1; F := ifactors(n)[2]; for f in F do if not divides(f[1]-1, n) then
%p A300330 P := P*f[1]^f[2] fi od; P end: seq(A300330(n), n=1..100);
%t A300330 a[n_]:=If[OddQ[n], 1, Denominator[BernoulliB[n]/n]/Denominator[BernoulliB[n]]]; Table[n/a[n], {n, 1, 100}] (* _Vincenzo Librandi_, Mar 12 2018 *)
%o A300330 (Julia)
%o A300330 using Nemo
%o A300330 function A300330(n) P = 1
%o A300330     for (p, e) in factor(ZZ(n))
%o A300330         ! divisible(ZZ(n), p - 1) && (P *= p^e) end
%o A300330 P end
%o A300330 [A300330(n) for n in 1:72] |> println
%o A300330 (Magma) [n/(Denominator(Bernoulli(n)/n)/Denominator(Bernoulli(n))): n in [1..100]]; // _Vincenzo Librandi_, Mar 12 2018
%Y A300330 Cf. A193267, A195989, A300711.
%K A300330 nonn
%O A300330 1,3
%A A300330 _Peter Luschny_, Mar 12 2018