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A355769
Numbers k such that both k and k+1 can be written as the sum of two nonzero squares.
0
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
1,1
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.
EXAMPLE
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.
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Angad Singh, Jul 16 2022
STATUS
approved