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 A020757 Numbers that are not the sum of two triangular numbers. 19
 5, 8, 14, 17, 19, 23, 26, 32, 33, 35, 40, 41, 44, 47, 50, 52, 53, 54, 59, 62, 63, 68, 71, 74, 75, 77, 80, 82, 85, 86, 89, 95, 96, 98, 103, 104, 107, 109, 113, 116, 117, 118, 122, 124, 125, 128, 129, 131, 134, 138, 140, 143, 145, 147, 149, 152, 155, 158, 161, 162, 166, 167 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS A052343(a(n)) = 0. - Reinhard Zumkeller, May 15 2006 Numbers of the form (p^(2k+1)s-1)/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 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 A nonnegative integer n is in this sequence if and only if A000729(n) = 0. - Michael Somos, Feb 13 2011 Terms of the form 4*a(n) + 1 are terms of A022544. - XU Pingya, Aug 05 2018 LINKS T. D. Noe, Table of n, a(n) for n = 1..10000 John A. Ewell, On Sums of Triangular Numbers and Sums of Squares, The American Mathematical Monthly, Vol. 99, No. 8 (October 1992), pp. 752-757. [From Ant King, Dec 02 2010] U. V. Satyanarayana, On the representation of numbers as sums of triangular numbers, The Mathematical Gazette, 45(351):40-43, February 1961. [From Hans J. H. Tuenter, Oct 11 2009] MATHEMATICA data = Reduce[m (m + 1) + n (n + 1) == 2 # && 0 <= m && 0 <= n, {m, n}, Integers] & /@ Range; Position[data, False] // Flatten  (* Ant King, Dec 05 2010 *) PROG (Haskell) a020757 n = a020757_list !! (n-1) a020757_list = filter ((== 0) . a052343) [0..] -- Reinhard Zumkeller, Jul 25 2014 CROSSREFS Complement of A020756. Cf. A052343. Sequence in context: A314473 A314474 A314475 * A314476 A314477 A314478 Adjacent sequences:  A020754 A020755 A020756 * A020758 A020759 A020760 KEYWORD nonn AUTHOR STATUS approved

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Last modified February 23 07:09 EST 2020. Contains 332159 sequences. (Running on oeis4.)