A sequence based on consecutive primes

[Nick Hobson]

On Mon, 07 May 2007 23:44:51 +0100, Don Reble <djr at nk.ca> wrote:

>>   a = sqrt((p^2 + q^2)/2 - 1).   (1)
>>
>> ...the smallest prime q > p + 2 which makes a an integer, ...
>> 0 is used if there is no such q:
>>
>> 11, 263, 59, 23, 101, 109, 0, 151, 193, 79, 269, 277, 311, 0, 179, 83,
>> 83003, 479, 487, 181, 563, 571, 613, 1201, 157, 141509, 739, 773, 479
>>
>> ... There is no such q for p = 19 and p = 47
>> (consider (1) modulo 7), but I can find no similar proof of
>> impossibility for p = 127, which would generate the next term in the
>> above sequence.
>
>     I get p=19,q=1278886952463697; p=127,q=6858037981;
>     but I find no q for p=47.
>     Even though I distrust the modulo-7 proof, how does it go?
>

Thanks Don.

You are correct, of course; the modulo 7 proof is flawed.  It was a  
(buggy) script that checked whether any values of a^2 are quadratic  
residues modulo various small primes.

Nick



Straightforward duplicates:

A056257 and A101135
A107749 and A126848
A100939 and A100948
A100940 and A100963
A056263 and A101156
A083222 and A083298
A088302 and A096073
A003523 and A018781
A104003 and A104033
A077519 and A078215
A073371 and A127977


Possible duplicates:

A108639 and A109975
http://www.research.att.com/~njas/sequences/?q=id:A108639|id:A109975

A080710 and A115837 (offsets are different)
http://www.research.att.com/~njas/sequences/?q=id:A080710|id:A115837

A038342 and A122596
http://www.research.att.com/~njas/sequences/?q=id:A038342|id:A122596

A037314 and A037457
http://www.research.att.com/~njas/sequences/?q=id:A037314|id:A037457


I think the title of A098037 doesn't accurately describe the sequence shown;
"Number of divisors with multiplicity of the sums of two consecutive
primes." I read "Number of divisors" to mean including non-prime as well as
prime divisors. For the example shown in the sequence definition, A098037(2)
= prime(2) + prime(3) = 3 + 5 = 8 = 2*2*2. Based on the title I believe 4
and 8 (and maybe 1?) should also be included, so A098037(2) >= 5 (not sure
if "with multiplicity" means that you'd also count another 2, as 4*2 = 8).
As it stands, A105049 is a duplicate of this sequence. Perhaps the title of
A098037 could be changed to "Number of prime divisors, counted with
multiplicity, of the sum of two consecutive primes". Here is a link for
comparison:

http://www.research.att.com/~njas/sequences/?q=id:A098037|id:A105049