# Number of positive solutions of the Diophantine x*p+y*q=(p+q)^3,

Max Alekseyev maxale at gmail.com
Tue Feb 26 14:11:33 CET 2008

```Zak,

I'm not sure if this sequence is of any interest (it looks rather
arbitrary to me) but I confirm that your terms are correct.
Also, I remark that there is a faster way to compute its terms, using
Bezout coefficients. This is a sample PARI/GP code:

{ a(n) = local(b,p,q); p=prime(n); q=prime(n+1); b=bezout(p,q);
m=(p+q)^3; m*b[1]\q + m*b[2]\p + 1 }

Regards,
Max

On Mon, Feb 25, 2008 at 7:26 PM, zak seidov <zakseidov at yahoo.com> wrote:
> Dear seqfans,
>  is it
>  new/correct/interesting/nice,
>  or
>  old/wrong/dumb?
>  thanks, zak
>
>  %I A000001
>  %S A000001
>  21,34,49,76,97,122,145,169,211,241,274,313,337,361,401,449,481,513,553,577,609,649,689,745,793,817,841,865,889,963,1033,1073,1105,1153,1201,1233,1281,1321,1361,1409,1441,1489,1537,1561,1585,1641,1737,1801,1825,1849,1889,1921,1969,2033,2081,2129,2161,2193,2233,2257,2305,2401,2473,2497,2521,2593,2673,2737,2785,2809,2849,2905,2961,3009,3049,3089,3145,3193,3241,3313,3361,3409,3457,3489,3529,3569,3625,3673,3697,3721,3785,3865,3913,3961,4009,4049,4121,4177,4257,4353
>  %N A000001 Number of positive solutions of the
>  Diophantine x*p+y*q=(p+q)^3, p=n-th prime, q=(n+1)-th
>  prime.
>  %e A000001 a(1)=21 because Diophantine 2x+3y=(2+3)^3
>  has 21 positive solutions {x,y}:
>  {1,41},{4,39},{7,37},{10,35},{13,33},{16,31},{19,29},{22,27},{25,25},{28,23},{31,21},{34,19},{37,17},{40,15},{43,13},{46,11},{49,9},{52,7},{55,5},{58,3},{61,1};
>  a(2)=34 because Diophantine 3x+5y=(3+5)^3 has 34
>  positive solutions {x,y}:
>  {4,100},{9,97},{14,94},{19,91},{24,88},{29,85},{34,82},{39,79},{44,76},{49,73},{54,70},{59,67},{64,64},{69,61},{74,58},{79,55},{84,52},{89,49},{94,46},{99,43},{104,40},{109,37},{114,34},{119,31},{124,28},{129,25},{134,22},{139,19},{144,16},{149,13},{154,10},{159,7},{164,4},{169,1}.
>  %t A000001
>  Table[With[{p=Prime[n],q=Prime[n+1]},Floor[(p+q)^2/p]+Floor[(p+q)^2/q]+1],{n,1,100}]
>  %O A000001 1
>  %K A000001 ,nonn,
>  %A A000001 Zak Seidov (zakseidov(AT)yahoo.com), Feb 25
>  2008
>
>
>
>
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>

```