%I
%S 5,8,14,17,19,23,26,32,33,35,40,41,44,47,50,52,53,54,59,62,63,68,71,
%T 74,75,77,80,82,85,86,89,95,96,98,103,104,107,109,113,116,117,118,122,
%U 124,125,128,129,131,134,138,140,143,145,147,149,152,155,158,161,162,166,167
%N Numbers that are not the sum of two triangular numbers.
%C A052343(a(n)) = 0.  _Reinhard Zumkeller_, May 15 2006
%C Numbers of the form (p^(2k+1)s1)/4, where p is a prime number of the form 4n+3, and s is a number of the form 4m+3 and prime to p, are not expressible as the sum of two triangular numbers. See Satyanarayana (1961), Theorem 2.  _Hans J. H. Tuenter_, Oct 11 2009
%C An integer n is in this sequence if and only if at least one 4k+3 prime factor in the canonical form of 4n+1 occurs with an odd exponent.  _Ant King_, Dec 02 2010
%C A nonnegative integer n is in this sequence if and only if A000729(n) = 0.  _Michael Somos_, Feb 13 2011
%C Terms of the form 4*a(n) + 1 are terms of A022544.  _XU Pingya_, Aug 05 2018
%H T. D. Noe, <a href="/A020757/b020757.txt">Table of n, a(n) for n = 1..10000</a>
%H John A. Ewell, <a href="http://www.jstor.org/stable/2324243">On Sums of Triangular Numbers and Sums of Squares</a>, The American Mathematical Monthly, Vol. 99, No. 8 (October 1992), pp. 752757. [From _Ant King_, Dec 02 2010]
%H U. V. Satyanarayana, <a href="http://www.jstor.org/stable/3614771">On the representation of numbers as sums of triangular numbers</a>, The Mathematical Gazette, 45(351):4043, February 1961. [From _Hans J. H. Tuenter_, Oct 11 2009]
%t data = Reduce[m (m + 1) + n (n + 1) == 2 # && 0 <= m && 0 <= n, {m, n}, Integers] & /@ Range[167]; Position[data, False] // Flatten (* _Ant King_, Dec 05 2010 *)
%o (Haskell)
%o a020757 n = a020757_list !! (n1)
%o a020757_list = filter ((== 0) . a052343) [0..]
%o  _Reinhard Zumkeller_, Jul 25 2014
%Y Complement of A020756.
%Y Cf. A052343.
%K nonn
%O 1,1
%A _David W. Wilson_
