A355769 Numbers k such that both k and k+1 can be written as the sum of two nonzero squares.

17, 25, 40, 52, 72, 73, 89, 97, 100, 116, 136, 145, 148, 169, 180, 193, 225, 232, 233, 241, 244, 260, 288, 289, 292, 305, 313, 337, 369, 388, 400, 404, 409, 424, 449, 457, 481, 520, 521, 544, 548, 577, 584, 585, 592, 612, 625, 628, 640, 656, 673, 676, 697, 724

Offset: 1

Author

Angad Singh, Jul 16 2022

Keywords

• nonn

Comments

The numbers in the sequence are useful in solving various second-degree Diophantine equations.

The identity (3n-12)^2 + (4n-12)^2 + 1 = (3n-8)^2 + (4n-15)^2 proves that there are infinitely many such numbers in this sequence.

Examples

			17 is a term since 17 = 4^2 + 1^2 and 17 + 1 = 18 = 3^2 + 3^2.
169 is a term since 169 = 5^2 + 12^2 and 169 + 1 = 170 = 1^2 + 13^2.
		

Crossrefs

Cf. A000290, A000404, A000415.

Programs

PARI

is1(n)= for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2)); \\ A000404
isok(k) = is1(k) && is1(k+1); \\ Michel Marcus, Jul 18 2022