Bernoulli numbers and arithmetic progressions

[Eric W. Weisstein]

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.