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

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, 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, 1, 2, 63, 64, 1, 6, 1, 4, 3, 10, 1, 72, 1, 2, 3, 4, 1, 78, 1, 80, 81, 2, 1, 84

Offset: 1

Author

Paul Curtz, Dec 21 2012

Keywords

• nonn

Comments

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.

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.

a(10*k) = 10, 20, 30, 40, 50, 60, 10, 70, 80, 90, 100,... for k= 1, 2, 3,....

Examples

			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.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Programs

Magma

[Denominator(Bernoulli(2*n)/n)/Denominator(Bernoulli(2*n)): n in [1..100]]; // Vincenzo Librandi, Mar 12 2018

Maple

A195989 := proc(n)
    q1 := denom(bernoulli(2*n)/n) ;
    q2 := denom(bernoulli(2*n)) ;
    q1/q2 ;
end proc: # R. J. Mathar, Jan 06 2013
# Alternatively, without Bernoulli numbers:
A195989 := proc(n) local P, F, f, divides; divides := (a,b) -> is(irem(b,a) = 0):
P := 1; F := ifactors(2*n)[2]; for f in F do if not divides(f[1]-1, 2*n) then
P := P*f[1]^f[2] fi od; n/P end: seq(A195989(n),n=1..84); # Peter Luschny, Mar 12 2018

Mathematica

a[n_] := Denominator[ BernoulliB[2*n]/n] / Denominator[ BernoulliB[2*n]]; Table[a[n], {n, 1, 84}] (* Jean-François Alcover, Jan 04 2013 *)

PARI

a(n) = my(b=bernfrac(2*n)); denominator(b/n)/denominator(b); \\ Michel Marcus, Mar 12 2018

Formula

a(n) = A193267(2*n)/2 = A036283(n) / A002445(n).

a(n) = n/A300711(n). - Peter Luschny, Mar 12 2018

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