A273236
Primes p such that p + k is the sum of two nonzero squares in exactly k ways for all k = 1, 2, 3.
Original entry on oeis.org
563047, 1186631, 1205647, 1421647, 1871503, 2058047, 2615047, 2739103, 2795047, 3703463, 3743647, 4106447, 4723847, 4748047, 4758847, 5744447, 6991847, 8376847, 9951047, 10014847, 12214303, 12773447, 14161183, 14402447, 15232031, 15630847
Offset: 1
The prime 563047 is a term because 563048 = 218^2 + 718^2, 563049 = 165^2 + 732^2 = 357^2 + 660^2 and 563050 = 71^2 + 747^2 = 141^2 + 737^2 = 505^2 + 555^2.
-
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) = isprime(n) && is(n+1, 1) && is(n+2, 2) && is(n+3, 3);
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.
Original entry on oeis.org
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
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.
-
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 *)
-
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);
Showing 1-2 of 2 results.
Comments