A073613 Triangular numbers which are the sum of two squares.

0, 1, 10, 36, 45, 136, 153, 325, 666, 820, 1225, 1378, 2080, 2628, 2701, 3240, 3321, 4005, 4753, 5050, 6786, 7381, 9316, 10440, 10585, 11026, 14365, 16290, 18721, 19306, 25425, 27028, 27261, 29161, 29890, 32896, 33930, 41616, 41905, 42778

Offset: 1

Author

Jason Earls, Aug 29 2002

Keywords

• easy
• nonn

Comments

The squares may be zero.

Examples

			0 = A000217(0) = A001481(1) = 0^2 + 0^2 is listed here as a(1).
1 = A000217(1) = A001481(2) = 1^2 + 0^2 is listed here as a(2).
10 = A000217(4) = A001481(8) = 1^2 + 9^2 is listed here as a(3).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000217 (triangular numbers), A001481 (sums of two squares).

Programs

Maple

filter:= proc(n)
  andmap(t -> (t[1] mod 4 <> 3 or t[2]::even), ifactors(n)[2])
end proc:
select(filter, [seq(i*(i+1)/2, i=0..500)]); # Robert Israel, Nov 22 2017

Mathematica

t = Range[0, 250]^2; t1 = Flatten[Table[a + b, {a, t}, {b, t}]]; t2 = Accumulate[Range[300]]; Intersection[t1, t2] (* Jayanta Basu, Jul 03 2013 *)
Select[Union[Total/@Tuples[Range[0,300]^2,2]],OddQ[Sqrt[8#+1]]&] (* Harvey P. Dale, Apr 22 2015 *)

PARI

is_A073613(n)=is_A000217(n)&&is_A001481(n) \\ M. F. Hasler, Nov 20 2017

Python

from itertools import count, islice
from sympy import factorint
def A073613_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(n).items()),(m*(m+1)//2 for m in count(0)))
A073613_list = list(islice(A073613_gen(),30)) # Chai Wah Wu, Jun 28 2022

Formula

Intersection of A000217 and A001481.

Extensions

Edited and initial 0 added by M. F. Hasler, Nov 20 2017