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A020757
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Numbers that are not the sum of two triangular numbers.
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19
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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
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OFFSET
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1,1
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COMMENTS
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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
4*a(n) + 1 are terms of A022544. - XU Pingya, Aug 05 2018 [Actually, k is here if and only if 4*k + 1 is in A022544. - Jianing Song, Feb 09 2021]
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LINKS
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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]
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
a020757 n = a020757_list !! (n-1)
a020757_list = filter ((== 0) . a052343) [0..]
-- Reinhard Zumkeller, Jul 25 2014
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CROSSREFS
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Complement of A020756.
Cf. A052343, A022544.
Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^m is zero: A090864 (m=1), A213250 (m=2), A014132 (m=3), A302056 (m=4), A302057 (m=5), this sequence (m=6), A322430 (m=8), A322431 (m=10), A322432 (m=14), A322043 (m=15), A322433 (m=26).
Sequence in context: A314473 A314474 A314475 * A314476 A314477 A314478
Adjacent sequences: A020754 A020755 A020756 * A020758 A020759 A020760
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KEYWORD
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nonn,changed
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AUTHOR
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David W. Wilson
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STATUS
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approved
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