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%I #16 May 22 2016 00:34:10
%S 3444,25456,35860,55544,78936,79740,93660,102612,110676,116788,122512,
%T 131808,145680,182624,184936,194184,235848,263988,267060,270480,
%U 273740,277416,284352,294756,305160,308676,343356,353760,360696,384924,410404,416136,465844
%N 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.
%C Sequence lists square roots of square terms of A273318.
%C 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, ...
%e 3444 is a term because;
%e 3444^2 = 756^2 + 3360^2.
%e 3444^2 + 1 = 681^2 + 3376^2 = 1^2 + 3444^2.
%e 3444^2 + 2 = 83^2 + 3443^2 = 1547^2 + 3077^2 = 1987^2 + 2813^2.
%t 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 *)
%o (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; }
%o isok(n) = is(n^2, 1) && is(n^2+1, 2) && is(n^2+2, 3);
%Y Cf. A273318.
%K nonn
%O 1,1
%A _Altug Alkan_, May 20 2016
%E a(7)-a(33) from _Giovanni Resta_, May 20 2016
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