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

1,3

Comments

The squares may be zero.

Links

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

Formula

Intersection of A000217 and A001481.

Example

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).

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A343155 A223305 A176575 * A346386 A117404 A359959

Adjacent sequences: A073610 A073611 A073612 * A073614 A073615 A073616

Keyword

easy,nonn

Author

Jason Earls, Aug 29 2002

Extensions

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

Status

approved