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 A074885 Numbers not of the form x^2 + M*y^2 for integers x > 0, y > 1, M > 0. 2
 1, 2, 3, 4, 6, 7, 11, 14, 15, 23, 30, 35, 38, 39, 42, 47, 62, 71, 78, 83, 87, 95, 110, 119, 138, 143, 155, 158, 167, 182, 195, 203, 210, 215, 222, 227, 230, 255, 263, 282, 287, 302, 318, 327, 335, 383, 390, 395, 398, 435, 447, 455, 462, 483, 503 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS "... 730847, 911027, 1011218, 1122558, 1153547, 1191302, 1195862, 1198823, 1200023, 1215843, 1230990, 1586343, 1607627, 1875902. There is strong numerical evidence that the series ends at 1875902. I calculate that the series has 436 members <= 1875902. 1875902 looks like the largest natural number n with this property (I have checked up to 100,000,000). If true, every sufficiently large number is expressible as x^2 + M*y^2 with x > 0, y > 1, M > 0), or (prosaically) as the sum of a square (A000290) and a not squarefree number (A013929)." - Chris Boyd (and modified by Robert G. Wilson v, Sep 30 2012) Number of terms less than or equal to 10^k, k=1...7: 6, 22, 81, 210, 367, 424, 436. Least number of the form x^2 + m*y^2 in k different ways, k=0...: 1, 5, 21, 13, 25, 37, 41, 68, 52, 81, 73, 100, 97, 160, 169, 148, 145, 153, 193, 261, 288, ..., . - Robert G. Wilson v, Sep 30 2012 REFERENCES Postings to sci.math Aug 24 2002, 03:04PM and Aug 26 2002, 03:07PM by Chris Boyd LINKS T. D. Noe, Table of n, a(n) for n = 1..436 (probably complete) Eric Weisstein's World of Mathematics, Pell Equation. MATHEMATICA notOfTheFormQ[n_] := Do[r = Reduce[x >= 1 && y > 1 && x^2 + m*y^2 == n, {x, y}, Integers]; If[r =!= False, Return[True]], {m, 1, Ceiling[(n - 1)/4]}] =!= True; Reap[ Do[ If[ notOfTheFormQ[n], Print[n]; Sow[n]], {n, 1, 600}]][[2, 1]] (* Jean-François Alcover, Sep 28 2012 *) fQ[n_] := Block[{flg = 0, lmt = 1 + Floor@ Sqrt@ n, m, x, y = 2}, While[y < lmt && flg == 0, x = 1; While[m = Floor[(n - x^2)/y^2]; m > 0 && Mod[n - x^2, y^2] != 0, x++]; If[n == x^2 + m*y^2, flg = 1]; y++]; flg == 0]; Select[Range@509, fQ] (* much faster than the above, Robert G. Wilson v, Sep 29 2012 *) mx = 1000; cnt = Table[0, {mx}]; Do[q = x^2 + m*y^2; If[q <= mx, cnt[[q]]++], {m, (mx - 1)/4}, {x, Sqrt[mx - 4 m]}, {y, 2, Sqrt[(mx - x^2)/m]}]; Flatten[Position[cnt, 0]] (* T. D. Noe, Sep 28 2012 *) PROG (Perl) \$xleast=1; \$yleast=2; \$start=1; \$range=2000000; \$xprev=\$xleast-1; for (\$k=0; (\$k+\$yleast)*(\$k+\$yleast) <= \$start+\$range; \$k++ ) { \$asquares[\$k]=(\$k+\$yleast)*(\$k+\$yleast); } for (\$k=\$start; \$k<\$start+\$range; \$k++ ) {&donum; } sub donum { \$nm=\$k-\$xprev*\$xprev; for (\$i=\$xprev; \$i<=\$nm; \$i++ ) {if ( (\$nm -= \$i+\$i+1) < 0 ) { last; } if (&ncheck(\$nm) > 0) { return; } } printf("%d\n", \$k); } sub ncheck { \$j=0; while ( (\$sq = \$asquares[\$j++ ]) <= \$_[0] ) { if (\$_[0] % \$sq == 0) { return 1; } } return 0; } (PARI) isOK(n) = my(k=1); while(k*k

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Last modified December 9 08:41 EST 2021. Contains 349627 sequences. (Running on oeis4.)