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A020757 Numbers that are not the sum of two triangular numbers. 10
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. [From 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.[Added by 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

REFERENCES

Ewell, John A.; On Sums of Triangular Numbers and Sums of Squares, The American Mathematical Monthly, Vol. 99, No. 8,  (October 1992), pp. 752-757. [Added by 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]

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

MATHEMATICA

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*)

CROSSREFS

Complement of A020756.

Sequence in context: A020736 A217185 A160421 * A049693 A084139 A092590

Adjacent sequences:  A020754 A020755 A020756 * A020758 A020759 A020760

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved

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Last modified April 21 02:07 EDT 2014. Contains 240824 sequences.