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A073613 Triangular numbers which are the sum of two squares.

%I A073613 #26 Jun 28 2022 19:10:52
%S A073613 0,1,10,36,45,136,153,325,666,820,1225,1378,2080,2628,2701,3240,3321,
%T A073613 4005,4753,5050,6786,7381,9316,10440,10585,11026,14365,16290,18721,
%U A073613 19306,25425,27028,27261,29161,29890,32896,33930,41616,41905,42778
%N A073613 Triangular numbers which are the sum of two squares.
%C A073613 The squares may be zero.
%H A073613 Robert Israel, <a href="/A073613/b073613.txt">Table of n, a(n) for n = 1..10000</a>
%F A073613 Intersection of A000217 and A001481.
%e A073613 0 = A000217(0) = A001481(1) = 0^2 + 0^2 is listed here as a(1).
%e A073613 1 = A000217(1) = A001481(2) = 1^2 + 0^2 is listed here as a(2).
%e A073613 10 = A000217(4) = A001481(8) = 1^2 + 9^2 is listed here as a(3).
%p A073613 filter:= proc(n)
%p A073613   andmap(t -> (t[1] mod 4 <> 3 or t[2]::even), ifactors(n)[2])
%p A073613 end proc:
%p A073613 select(filter, [seq(i*(i+1)/2, i=0..500)]); # _Robert Israel_, Nov 22 2017
%t A073613 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 *)
%t A073613 Select[Union[Total/@Tuples[Range[0,300]^2,2]],OddQ[Sqrt[8#+1]]&] (* _Harvey P. Dale_, Apr 22 2015 *)
%o A073613 (PARI) is_A073613(n)=is_A000217(n)&&is_A001481(n) \\ _M. F. Hasler_, Nov 20 2017
%o A073613 (Python)
%o A073613 from itertools import count, islice
%o A073613 from sympy import factorint
%o A073613 def A073613_gen(): # generator of terms
%o A073613     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)))
%o A073613 A073613_list = list(islice(A073613_gen(),30)) # _Chai Wah Wu_, Jun 28 2022
%Y A073613 Cf. A000217 (triangular numbers), A001481 (sums of two squares).
%K A073613 easy,nonn
%O A073613 1,3
%A A073613 _Jason Earls_, Aug 29 2002
%E A073613 Edited and initial 0 added by _M. F. Hasler_, Nov 20 2017