# Bernoulli numbers and arithmetic progressions

cino hilliard hillcino368 at hotmail.com
Wed Feb 4 20:14:23 CET 2004

>From: "Eric W. Weisstein" <eww at wolfram.com>
>To: Hans Havermann <hahaj at rogers.com>
>CC: seqfan at ext.jussieu.fr
>Subject: Re: Bernoulli numbers and arithmetic progressions
>Date: Wed, 4 Feb 2004 08:12:12 -0600 (CST)
>
Hi Eric,
The pari code below produces output that agrees with yours.

I am running the pari from 20000 to 30000 since you are going from 10000 -
?.
We may be able to get some speed improvement if someone knows how to program
the expression into Xavier Gourdon's Pifast hypergometric series solver. Is
Pascal Sebah on this list?
This would enable a community effort with assigned blocks of numbers.

(18:29) gp > bern2(19,5236,10000)
A(19) = 5236,37                              This term was repeated from
prior calc to test.
A(20) = 5862,617
A(21) = 5902,37
A(22) = 6227,593
A(23) = 6332,67
A(24) = 6402,59
A(25) = 6438,103
A(26) = 6568,37
A(27) = 7234,37
A(28) = 7900,37
A(29) = 8113,59
A(30) = 8434,101
A(31) = 8543,67
A(32) = 8557,157
A(33) = 8566,37
A(34) = 9232,37
A(35) = 9611,149
A(36) = 9670,233
A(37) = 9824,59
A(38) = 9891,131
A(39) = 9898,37
time = 8h, 46mn, 29,562 ms.

bern2(c,m1,m2) =
{
for(n=m1,m2,
n2=n+n;
a = numerator(bernfrac(n2)/(n2));           \\ A001067
b = numerator(bernfrac(n2)/(n2*(n2-1)));    \\ A046968
if(a <> b,print("A("c") = "n","a/b);c++)
)
}

Cino

>On Wed, 4 Feb 2004, Hans Havermann wrote:
>
> > On Feb 4, 2004, at 6:55 AM, Roland Bacher wrote:
> >
> > > all values yielding 37 are of the form: 574+666*k, k=0,1,2,3,4,...
> > > and form thus an arithmetic progression with step
> > > 666=18*37=((37-1)/2)*37;
> >
> > Very vice.
> >
> > Here are the data points (<10000):
> >
> > {574, 37}, {1185, 103}, {1240, 37}, {1269, 59}, {1376, 131}, {1906,
> > 37}, {1910, 67}, {2572, 37}, {2689, 283}, {2980, 59}, {3238, 37},
> > {3384, 101}, {3801, 691}, {3904, 37}, {4121, 67}, {4570, 37}, {4691,
> > 59}, {4789, 157}, {5236, 37}, {5862, 617}, {5902, 37}, {6227, 593},
> > {6332, 67}, {6402, 59}, {6438, 103}, {6568, 37}, {7234, 37}, {7900,
> > 37}, {8113, 59}, {8434, 101}, {8543, 67}, {8557, 157}, {8566, 37},
> > {9232, 37}, {9611, 149}, {9670, 233}, {9824, 59}, {9891, 131}, {9898,
> > 37}
>
>Very nice.  I have nothing else to add except to verify the above numbers
>and note that there are no others <= 10240 (and counting).
>
>574, 1185, 1240, 1269, 1376, 1906, 1910, 2572, 2689, 2980, 3238, 3384,
>3801, 3904, 4121, 4570, 4691, 4789, 5236, 5862, 5902, 6227, 6332, 6402,
>6438, 6568, 7234, 7900, 8113, 8434, 8543, 8557, 8566, 9232, 9611, 9670,
>9824, 9891, 9898
>
>37, 103, 37, 59, 131, 37, 67, 37, 283, 59, 37, 101, 691, 37, 67, 37, 59,
>157, 37, 617, 37, 593, 67, 59, 103, 37, 37, 37, 59, 101, 67, 157, 37, 37,
>149, 233, 59, 131, 37
>
>Cheers,
>-E
>
> > So, empirically, not one miss for 37.
>
>

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