%I
%S 0,1,2,3,4,5,7,8,9,10,13,15,16,17,21,24,27,28,29,32,33,34,42,49,54,59,
%T 66,68,72,79,80,81,85,86,95,96,99,102
%N Numbers n such that n! is the sum of two triangular numbers.
%C Numbers z such that there are nonnegative numbers x and y such that x(x+1)/2 + y(y+1)/2 = z!. Equivalently, (2x+1)^2+(2y+1)^2 = 8z!+2. A necessary and sufficient condition for this is that all the prime factors of 4z!+1 that are congruent to 3 mod 4 occur to even powers (cf. A001481).
%C Based on an email from _R. K. Guy_ to the Sequence Fans Mailing List, Sep 10 2010.
%C See A152089 for further links.
%H Factor Database, <a href="http://factordb.com/index.php?query=4*z%21%2B1&perpage=200">Factors of the numbers 4z!+1</a>
%e 0!=1!=T(0)+T(1), 2!=T(1)+T(1), 3!=T(0)+T(2), 4!=T(2)+T(4), 5!=T(5)=T(14), 7!=T(45)+T(89), 8!=T(89)+T269), 9!=T(210)+T(825), 10!=T(760)+T(2610), 13!= T(71504)+T(85680), 15!=T(213384)+T(1603064), etc.
%t triQ[n_] := IntegerQ@ Sqrt[8 n + 1]; fQ[n_] := Block[{k = 1, lmt = Floor@Sqrt[2*n! ], nf = n!}, While[k < lmt && ! triQ[nf  k (k + 1)/2], k++ ]; r = (Sqrt[8*(nf  k (k + 1)/2) + 1]  1)/2; Print[{k, r, n}]; If[IntegerQ@r, True, False]]; k = 1; lst = {}; While[k < 69, If[ fQ@ k, AppendTo[lst, k]]; k++ ]; lst
%Y A171099 gives the number of solutions. Cf. A000142, A000217, A001481, A076680. Complement of A152089.
%K nonn,more
%O 1,3
%A _Robert G. Wilson v_, Sep 10 2010
%E Edited by _N. J. A. Sloane_, Sep 24 2010
%E 69 eliminated by _N. J. A. Sloane_, Sep 24 2010 (see A152089).
%E Extended by G. Guninski and _D. S. McNeil_, Sep 24 2010
%E 95, 96, 99, 102 from G. Guninski, Oct 12 2010
