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

1,2

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,....

Links

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

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

Example

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.

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 *)

Prog

(PARI) a(n) = my(b=bernfrac(2*n)); denominator(b/n)/denominator(b); \\ Michel Marcus, Mar 12 2018
(Magma) [Denominator(Bernoulli(2*n)/n)/Denominator(Bernoulli(2*n)): n in [1..100]]; // Vincenzo Librandi, Mar 12 2018

Crossrefs

Sequence in context: A366283 A065331 A066262 * A174715 A069817 A071439

Adjacent sequences: A195986 A195987 A195988 * A195990 A195991 A195992

Keyword

nonn

Author

Paul Curtz, Dec 21 2012

Status

approved