cp's OEIS Frontend

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

%I A020757 #63 Aug 08 2024 01:45:23
%S A020757 5,8,14,17,19,23,26,32,33,35,40,41,44,47,50,52,53,54,59,62,63,68,71,
%T A020757 74,75,77,80,82,85,86,89,95,96,98,103,104,107,109,113,116,117,118,122,
%U A020757 124,125,128,129,131,134,138,140,143,145,147,149,152,155,158,161,162,166,167
%N A020757 Numbers that are not the sum of two triangular numbers.
%C A020757 A052343(a(n)) = 0. - _Reinhard Zumkeller_, May 15 2006
%C A020757 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
%C A020757 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
%C A020757 A nonnegative integer n is in this sequence if and only if A000729(n) = 0. - _Michael Somos_, Feb 13 2011
%C A020757 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]
%C A020757 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
%H A020757 T. D. Noe, <a href="/A020757/b020757.txt">Table of n, a(n) for n = 1..10000</a>
%H A020757 John A. Ewell, <a href="http://www.jstor.org/stable/2324243">On Sums of Triangular Numbers and Sums of Squares</a>, The American Mathematical Monthly, Vol. 99, No. 8 (October 1992), pp. 752-757. [From _Ant King_, Dec 02 2010]
%H A020757 U. V. Satyanarayana, <a href="http://www.jstor.org/stable/3614771">On the representation of numbers as sums of triangular numbers</a>, The Mathematical Gazette, 45(351):40-43, February 1961. [From _Hans J. H. Tuenter_, Oct 11 2009]
%e A020757 3 = 0 + 3 and 7 = 1 + 6 are not terms, but 8 = 1 + 1 + 6 is a term.
%t A020757 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 A020757 t = Array[PolygonalNumber, 18, 0]; Complement[Range@ 169, Flatten[ Outer[ Plus, t, t]]] (* _Robert G. Wilson v_, Aug 07 2024 *)
%o A020757 (Haskell)
%o A020757 a020757 n = a020757_list !! (n-1)
%o A020757 a020757_list = filter ((== 0) . a052343) [0..]
%o A020757 -- _Reinhard Zumkeller_, Jul 25 2014
%o A020757 (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
%Y A020757 Complement of A020756.
%Y A020757 Cf. A000217, A052343, A022544, A061336.
%Y A020757 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).
%K A020757 nonn
%O A020757 1,1
%A A020757 _David W. Wilson_