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A020757
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Numbers that are not the sum of two triangular numbers.
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21
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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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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
Integers m such that the smallest number of triangular numbers which sum to m is 3, hence A061336(a(n)) = 3. - Bernard Schott, Jul 21 2022
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LINKS
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EXAMPLE
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3 = 0 + 3 and 7 = 1 + 6 are not terms, but 8 = 1 + 1 + 6 is a term.
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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 *)
t = Array[PolygonalNumber, 18, 0]; Complement[Range@ 169, Flatten[ Outer[ Plus, t, t]]] (* Robert G. Wilson v, Aug 07 2024 *)
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PROG
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(Haskell)
a020757 n = a020757_list !! (n-1)
a020757_list = filter ((== 0) . a052343) [0..]
(PARI) is(n)=my(m9=n%9, f); if(m9==5 || m9==8, return(1)); f=factor(4*n+1); for(i=1, #f~, if(f[i, 1]%4==3 && f[i, 2]%2, return(1))); 0 \\ Charles R Greathouse IV, Mar 17 2022
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CROSSREFS
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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).
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KEYWORD
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nonn,changed
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AUTHOR
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STATUS
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approved
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