A020757 Numbers that are not the sum of two triangular numbers.

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

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

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]

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

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]

Example

3 = 0 + 3 and 7 = 1 + 6 are not terms, but 8 = 1 + 1 + 6 is a term.

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 *)

Prog

(Haskell)
a020757 n = a020757_list !! (n-1)
a020757_list = filter ((== 0) . a052343) [0..]
-- Reinhard Zumkeller, Jul 25 2014
(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

Crossrefs

Complement of A020756.

Cf. A000217, A052343, A022544, A061336.

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 * A361009 A314476 A314477

Adjacent sequences: A020754 A020755 A020756 * A020758 A020759 A020760

Keyword

nonn

Author

David W. Wilson

Status

approved