[seqfan] hello, Ramanujan and Bernoulli numbers

[Jud McCranie]

>
>
> Hello, I'm new to the SeqFan mailing list, but I've been a fan of=
 sequences
> for a long time.=A0 My interest in sequences are mostly in prime numbers=
 and
> number theory functions, mainly when computers are involved.
>
> I've contributed or extended several sequences.=A0 I want to mention two
that I
> just added (A33562 and 33563).=A0 I was looking at "The man who knew
infinity -
> A life of the genius Ramanujan, by Robert Kanigel.=A0 On page 91-92, it=
 says
> that Ramanujan stated (without proof!) that if take Bernoulli number B(n)
and
> reduce B(n)/n to lowest terms, the numerator is never a composite number.=
=A0
> The conjecture fails at the 11th non-zero B(n)!=A0=20
>
> I've been known to make conjectures on little data, but since the first
> several terms of this sequence are trivial, he made the conjecture on at
most
> 3 non-trivial examples!=A0 The numerators of the even Bernoulli numbers=
 are
> 1,1,1,1,1,5,691,7,3617,43867,174611 (A4039) and the numerators after
reducing
> by n are 1,1,1,1,1,691,1,3617,43867,174611 (A33562).=A0 But
174611=3D283*617.=A0 It
> seems to me to be a bold or risky statement to look at
> 1,1,1,1,1,691,1,3617,43867 and state that there are no composites in the
> sequence.
>
> A33562 gives the numerators of B(2n)/2n.=A0 A33563 gives the primes in the
> sequence, showing that Ramanujan's conjecture was usually wrong.



+---------------------------------------------------------+
| Jud McCranie jud.mccranie at mindspring.com or @camcat.com |
|                                                         |
| "We should regard the digital computer system as        |
| an instrument to assist the number theorist in          |
| investigating the properties of his universe -          |
| the natural numbers."  D. H. Lehmer, 1974 (paraphrased) |
+---------------------------------------------------------+