A191765 Integers that are a sum of two nonzero triangular numbers and also the sum of two nonzero square numbers.

2, 13, 18, 20, 25, 29, 34, 37, 58, 61, 65, 72, 73, 90, 97, 100, 101, 106, 130, 136, 137, 146, 148, 157, 160, 164, 169, 181, 193, 200, 202, 205, 208, 218, 225, 226, 232, 234, 241, 244, 245, 265, 272, 274, 277, 281, 288, 289, 298, 306, 328, 340, 346, 353, 370, 373, 388, 389, 400

Offset: 1

Author

Ant King, Jun 22 2011

Keywords

• nonn
• easy

Comments

A134422 is a subsequence. - Franklin T. Adams-Watters, Jun 25 2011

Examples

			25 is the sum of two nonzero triangular numbers: 10 + 15, and of two nonzero squares: 9 + 16; so 25 is in the sequence.
9 is the sum of two nonzero triangular numbers: 3 + 6, but can be represented as the sum of two squares only using zero: 0 + 9; so 9 is not in the sequence.
		

Links

• P. A. Piza, Problems for Solution: 4425, The American Mathematical Monthly, Vol. 58, No. 2, (February 1951), p. 113.
• P. A. Piza, G. W. Walker and C. M. Sandwick, Sr, Problem 4425, The American Mathematical Monthly, Vol. 59, No. 6, (June - July 1952), pp. 417-419.

Crossrefs

Cf. A000217, A000290, A191766, intersection of A000404 and A051533, A134422.

Programs

Mathematica

data=Length[Reduce[a^2+b^2==1/2 c (c+1)+1/2 d(d+1)== # && a>0 && b>0 && c>0 && d>0,{a,b,c,d},Integers]] &/@Range[400];DeleteCases[Table[If[data[[k]]>0,k,0],{k,1,Length[data]}],0]