A073613 - OEIS

%I #26 Jun 28 2022 19:10:52

%S 0,1,10,36,45,136,153,325,666,820,1225,1378,2080,2628,2701,3240,3321,

%T 4005,4753,5050,6786,7381,9316,10440,10585,11026,14365,16290,18721,

%U 19306,25425,27028,27261,29161,29890,32896,33930,41616,41905,42778

%N Triangular numbers which are the sum of two squares.

%C The squares may be zero.

%H Robert Israel, <a href="/A073613/b073613.txt">Table of n, a(n) for n = 1..10000</a>

%F Intersection of A000217 and A001481.

%e 0 = A000217(0) = A001481(1) = 0^2 + 0^2 is listed here as a(1).

%e 1 = A000217(1) = A001481(2) = 1^2 + 0^2 is listed here as a(2).

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

%p filter:= proc(n)

%p   andmap(t -> (t[1] mod 4 <> 3 or t[2]::even), ifactors(n)[2])

%p end proc:

%p select(filter, [seq(i*(i+1)/2, i=0..500)]); # _Robert Israel_, Nov 22 2017

%t 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 Select[Union[Total/@Tuples[Range[0,300]^2,2]],OddQ[Sqrt[8#+1]]&] (* _Harvey P. Dale_, Apr 22 2015 *)

%o (PARI) is_A073613(n)=is_A000217(n)&&is_A001481(n) \\ _M. F. Hasler_, Nov 20 2017

%o (Python)

%o from itertools import count, islice

%o from sympy import factorint

%o def A073613_gen(): # generator of terms

%o     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_list = list(islice(A073613_gen(),30)) # _Chai Wah Wu_, Jun 28 2022

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

%K easy,nonn

%O 1,3

%A _Jason Earls_, Aug 29 2002

%E Edited and initial 0 added by _M. F. Hasler_, Nov 20 2017