Bernoulli numbers
Robert G. Wilson v
rgwv at rgwv.com
Tue Feb 3 21:52:01 CET 2004
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, 2500}] .
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
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/
>
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