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%I #34 Jun 19 2017 00:32:59
%S 1,2,3,4,6,7,11,14,15,23,30,35,38,39,42,47,62,71,78,83,87,95,110,119,
%T 138,143,155,158,167,182,195,203,210,215,222,227,230,255,263,282,287,
%U 302,318,327,335,383,390,395,398,435,447,455,462,483,503
%N Numbers not of the form x^2 + M*y^2 for integers x > 0, y > 1, M > 0.
%C "... 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)
%C Number of terms less than or equal to 10^k, k=1...7: 6, 22, 81, 210, 367, 424, 436.
%C 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
%D Postings to sci.math Aug 24 2002, 03:04PM and Aug 26 2002, 03:07PM by _Chris Boyd_
%H T. D. Noe, <a href="/A074885/b074885.txt">Table of n, a(n) for n = 1..436</a> (probably complete)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PellEquation.html">Pell Equation.</a>
%t 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 *)
%t 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 *)
%t 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 *)
%o (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;}
%o (PARI) isOK(n) = my(k=1); while(k*k<n, if(!issquarefree(n-k^2), return(0)); k++); 1 s=[]; for(n=1, 1000, if(isOK(n), s=concat(s, n))); s \\ _Colin Barker_, Apr 26 2014
%K nonn,nice
%O 1,2
%A _Chris Boyd_ and _Robert G. Wilson v_, Sep 12 2002
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