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A073613
Triangular numbers which are the sum of two squares.
3
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
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
KEYWORD
easy,nonn
AUTHOR
Jason Earls, Aug 29 2002
EXTENSIONS
Edited and initial 0 added by M. F. Hasler, Nov 20 2017
STATUS
approved