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A193267 The number 1 alternating with the numbers A006953/A002445 (which are integers).

%I A193267 #36 Sep 08 2022 08:45:58
%S A193267 1,2,1,4,1,6,1,8,1,2,1,12,1,2,1,16,1,18,1,20,1,2,1,24,1,2,1,4,1,6,1,
%T A193267 32,1,2,1,36,1,2,1,40,1,42,1,4,1,2,1,48,1,2,1,4,1,54,1,8,1,2,1,60,1,2,
%U A193267 1,64,1,6,1,4,1,2,1,72,1,2,1,4,1,6,1,80,1,2,1,84,1,2,1,8,1,18,1,4,1,2,1,96,1,2,1,100
%N A193267 The number 1 alternating with the numbers A006953/A002445 (which are integers).
%C A193267 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 divides n. - _Peter Luschny_, Mar 12 2018
%H A193267 Vincenzo Librandi, <a href="/A193267/b193267.txt">Table of n, a(n) for n = 1..2000</a>
%F A193267 a(n+1) = A185633(n+1)/A027760(n+1).
%F A193267 a(n+1) = c(n+2)/c(n+1).
%p A193267 with(numtheory); a := proc(n) divisors(n); map(i->i+1, %); select(isprime, %);
%p A193267 mul(k^padic[ordp](n,k),k=%) end: seq(a(n), n=1..100); # _Peter Luschny_, Mar 12 2018
%p A193267 # Alternatively:
%p A193267 A193267 := proc(n) local P, F, f, divides; divides := (a,b) -> is(irem(b,a) = 0):
%p A193267 P := 1; F := ifactors(n)[2]; for f in F do if divides(f[1]-1, n) then
%p A193267 P := P*f[1]^f[2] fi od; P end: seq(A193267(n), n=1..100); # _Peter Luschny_, Mar 12 2018
%t A193267 a[n_] := If[OddQ[n], 1, Denominator[ BernoulliB[n]/n ] / Denominator[ BernoulliB[n]] ]; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Dec 21 2012 *)
%o A193267 (Magma) [Denominator(Bernoulli(n)/n)/Denominator(Bernoulli(n)): n in [1..100]]; // _Vincenzo Librandi_, Mar 12 2018
%o A193267 (Julia)
%o A193267 using Nemo
%o A193267 function A193267(n) P = 1
%o A193267     for (p, e) in factor(ZZ(n))
%o A193267         divisible(ZZ(n), p - 1) && (P *= p^e) end
%o A193267 P end
%o A193267 [A193267(n) for n in 1:100] |> println # _Peter Luschny_, Mar 12 2018
%K A193267 nonn
%O A193267 1,2
%A A193267 _Paul Curtz_, Dec 20 2012