A274971 Numbers k such that (x+1)^3 - x^3 = k*y^2 has integer solutions.

1, 7, 19, 31, 37, 43, 61, 67, 79, 91, 103, 127, 139, 151, 157, 163, 169, 199, 211, 217, 223, 247, 271, 283, 307, 313, 331, 343, 349, 367, 373, 379, 397, 403, 427, 439, 463, 469, 487, 499, 511, 523, 547, 553, 571, 577, 607, 613, 619, 631, 643, 661, 679, 691

Offset: 1

Author

Colin Barker, Jul 13 2016

Keywords

• nonn

Examples

			7 is in the sequence because, for instance, (167^3-166^3)/7 = 11881 = 109^2.
		

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000
• Dario A. Alpern, Quadratic two integer variable equation solver

Crossrefs

Cf. A001921 (k=1), A144929 (k=7), A145124 (k=19), A145323 (k=31), A145700 (k=37), A145336 (k=43), A274972 (k=61), A145212 (k=67), A145309 (k=79), A145530 (k=91), A147530 (k=103), A145720 (k=127).

Cf. A003215 is a subsequence; A004611 contains this sequence.

Programs

Mathematica

A004611=Select[Range[500],And@@(Mod[#,3]==1&)/@(First/@FactorInteger[#])&]; Select[A004611,Reduce[x^2+3== 12*#*y^2,{x,y},Integers]=!=False &] (* Ray Chandler, Jul 24 2016 *)

Extensions

More terms using solver at Alpern link by Ray Chandler, Jul 23 2016