A273341 Numbers n such that n^2+k-1 is the sum of two nonzero squares in exactly k ways for all k = 1, 2, 3.

3444, 25456, 35860, 55544, 78936, 79740, 93660, 102612, 110676, 116788, 122512, 131808, 145680, 182624, 184936, 194184, 235848, 263988, 267060, 270480, 273740, 277416, 284352, 294756, 305160, 308676, 343356, 353760, 360696, 384924, 410404, 416136, 465844

Offset: 1

Author

Altug Alkan, May 20 2016

Keywords

• nonn

Comments

Sequence lists square roots of square terms of A273318.

Numbers n such that (n+k-1)^2 is the sum of two nonzero squares in exactly k ways for all k = 1, 2, 3 are 11998, 40748, 54248, ...

Examples

			3444 is a term because;
3444^2 = 756^2 + 3360^2.
3444^2 + 1 = 681^2 + 3376^2 = 1^2 + 3444^2.
3444^2 + 2 = 83^2 + 3443^2 = 1547^2 + 3077^2 = 1987^2 + 2813^2.
		

Crossrefs

Cf. A273318.

Programs

Mathematica

nR[n_] := (SquaresR[2,n] + Plus @@ Pick[{-4,4}, IntegerQ /@ Sqrt[{n, n/2} ]])/8; Select[ Range[ 10^5], nR[#^2] == 1 && nR[#^2 + 1] == 2 && nR[#^2 + 2] == 3 &] (* Giovanni Resta, May 20 2016 *)

PARI

is(n, k) = {nb = 0; lim = sqrtint(n); for (x=1, lim, if ((n-x^2 >= x^2) && issquare(n-x^2), nb++); ); nb == k; }
isok(n) = is(n^2, 1) && is(n^2+1, 2) && is(n^2+2, 3);

Extensions

a(7)-a(33) from Giovanni Resta, May 20 2016