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A195989 Quotient of denominators of (BernoulliB(2n)/n) and BernoulliB(2n).

%I A195989 #41 Sep 08 2022 08:45:59
%S A195989 1,2,3,4,1,6,1,8,9,10,1,12,1,2,3,16,1,18,1,20,21,2,1,24,1,2,27,4,1,30,
%T A195989 1,32,3,2,1,36,1,2,3,40,1,42,1,4,9,2,1,48,1,50,3,4,1,54,11,8,3,2,1,60,
%U A195989 1,2,63,64,1,6,1,4,3,10,1,72,1,2,3,4,1,78,1,80,81,2,1,84
%N A195989 Quotient of denominators of (BernoulliB(2n)/n) and BernoulliB(2n).
%C A195989 The fixed points (entries equal to their index) are 1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 21, 24, 27, 30, 32, 36, 40, 42,... See A193267.
%C A195989 Are the indices of the 1's, that is 1, 5, 7, 11, 13,... , the sequence A069040 (checked to be true for their first 700 entries)? This provides another link between the Bernoulli numbers.
%C A195989 a(10*k) = 10, 20, 30, 40, 50, 60, 10, 70, 80, 90, 100,... for k= 1, 2, 3,....
%H A195989 Vincenzo Librandi, <a href="/A195989/b195989.txt">Table of n, a(n) for n = 1..2000</a>
%F A195989 a(n) = A193267(2*n)/2 = A036283(n) / A002445(n).
%F A195989 a(n) = n/A300711(n). - _Peter Luschny_, Mar 12 2018
%F A195989 2a(n) is the product over all prime powers p^e, where p^e is the highest power of p dividing 2n and p-1 divides 2n. - _Peter Luschny_, Mar 12 2018
%e A195989 a(1) = 6/6 =1, a(2) = 60/30 =2, a(3) =126/42 =3, a(4) = 120/30 =4, a(5) = 66/66 =1.
%p A195989 A195989 := proc(n)
%p A195989     q1 := denom(bernoulli(2*n)/n) ;
%p A195989     q2 := denom(bernoulli(2*n)) ;
%p A195989     q1/q2 ;
%p A195989 end proc: # _R. J. Mathar_, Jan 06 2013
%p A195989 # Alternatively, without Bernoulli numbers:
%p A195989 A195989 := proc(n) local P, F, f, divides; divides := (a,b) -> is(irem(b,a) = 0):
%p A195989 P := 1; F := ifactors(2*n)[2]; for f in F do if not divides(f[1]-1, 2*n) then
%p A195989 P := P*f[1]^f[2] fi od; n/P end: seq(A195989(n),n=1..84); # _Peter Luschny_, Mar 12 2018
%t A195989 a[n_] := Denominator[ BernoulliB[2*n]/n] / Denominator[ BernoulliB[2*n]]; Table[a[n], {n, 1, 84}] (* _Jean-François Alcover_, Jan 04 2013 *)
%o A195989 (PARI) a(n) = my(b=bernfrac(2*n)); denominator(b/n)/denominator(b); \\ _Michel Marcus_, Mar 12 2018
%o A195989 (Magma) [Denominator(Bernoulli(2*n)/n)/Denominator(Bernoulli(2*n)): n in [1..100]]; // _Vincenzo Librandi_, Mar 12 2018
%K A195989 nonn
%O A195989 1,2
%A A195989 _Paul Curtz_, Dec 21 2012