Bernoulli numbers

[Robert G. Wilson v]

Neil,

	Just supplied the Mathematica coding for the two sequences.
	
	I also set a[n_] := Numerator[BernoulliB[2n]/(2n)] and
                b[n_] := Numerator[BernoulliB[2n]/(2n(2n - 1))] and
Do[ If[ a[n] != b[n], Print[n, "  ", a[n]/b[n]]], {n, 1, 3000}] .
This produced the following results:

       n     a(n)/b(n)
	
	 574          37
	1185         103
         1240          37
	1269          59
	1376         131
	1906          37
	1910          67
	2572          37
	2689         283
	
	Does the ratio a(n)/b(n) always result in an Irregular prime?
        http://www.research.att.com/projects/OEIS?Anum=A000928

	

	Do I see two new sequences here? One titled 'n's for where A001067 differs
from A046968' and 'A001067(n)/A046968(n) for n's above.'

Thanx, Bob.


%I A001067
%S A001067 1,1,1,1,1,691,1,3617,43867,174611,77683,236364091,657931,3392780147,
%T A001067 1723168255201,7709321041217,151628697551,26315271553053477373,
%U A001067
154210205991661,261082718496449122051,1520097643918070802691,2530297234481911294093
%V A001067 1,-1,1,-1,1,-691,1,-3617,43867,-174611,77683,-236364091,657931,-3392780147,
%W A001067 1723168255201,-7709321041217,151628697551,-26315271553053477373,
%X A001067
154210205991661,-261082718496449122051,1520097643918070802691,-2530297234481911294093
%N A001067 Numerator of Bernoulli(2n)/(2n).
%C A001067 Also numerator of "modified Bernoulli number" b(2n) =
Bernoulli(2*n)/(2*n*n!). Denominators are in A057868.
...........
%t A001067 Table[ Numerator[ BernoulliB[2n]/(2n)], {n, 1, 22}] (from RGWv Feb 03 2004)
%K A001067 sign,frac,nice
%O A001067 1,6
%A A001067 njas, Richard E. Borcherds (reb(AT)math.berkeley.edu)


%I A046968
%S A046968 1,1,1,1,1,691,1,3617,43867,174611,77683,236364091,657931,3392780147,
%T A046968 1723168255201,7709321041217,151628697551,26315271553053477373,
%U A046968 154210205991661,261082718496449122051,1520097643918070802691
%V A046968 1,-1,1,-1,1,-691,1,-3617,43867,-174611,77683,-236364091,657931,-3392780147,
%W A046968
1723168255201,-7709321041217,151628697551,-26315271553053477373,154210205991661,
%X A046968 -261082718496449122051,1520097643918070802691
%N A046968 Numerators of coefficients in Stirling's expansion for ln Gamma(z).
...........
%t A046968 Table[ Numerator[ BernoulliB[2n]/(2n(2n - 1))], {n, 1, 22}] (from RGWv
Feb 03 2004)
%K A046968 frac,sign,nice
%O A046968 1,6
%A A046968 Douglas Stoll, dougstoll(AT)email.msn.com
%E A046968 More terms from Frank.Ellermann(AT)t-online.de, Jun 13 2001


N. J. A. Sloane wrote:

> Dear SeqFans,   Michael Somos made an interesting discovery.
> A001067 and A046968 are not the same!
> In fact: (speaking baby Mathematica talk, which is
> all that i know): 
> 
> a[n_] := Numerator[BernoulliB[2n]/(2n)]          (* A001067 *)
> b[n_] := Numerator[BernoulliB[2n]/(2n(2n-1))]    (* A046968 *)
> 
> For[n=1, n <= 580, n++, 
> If[ a[n] != b[n], Print[n, " ", a[n]/b[n]] ]
> ]
> 
> produces one line of output
> 
>                   574 37
> 
> In other words, the sequence of values of n such that
> A001067(n) differs from A046968(n) starts
> 574, ...
> and the associated ratios begin
> 37, ...
> 
> Could someone with more computing power than I have extend
> these two sequences?  PARI gives the same result.  Maple dies.
> 
> 
> Thanks
> 
> Neil
> 
> 
>  Neil J. A. Sloane
>  AT&T Shannon Labs, Room C233, 
>  180 Park Avenue, Florham Park, NJ 07932-0971
>  Email: njas at research.att.com
>  Office: 973 360 8415; fax: 973 360 8178
>  Home page: http://www.research.att.com/~njas/
>