A390113 - OEIS

%I #37 Nov 10 2025 20:06:29

%S 5,29,104,169,985,1364,2295,5741,33461,195025

%N Numbers i >= 1 such that 1/t_i + 1/t_j = 1/t_k, i > j > k, where t_r = r*(r+1)/2 is the r-th triangular number (A000217).

%C All terms of A001653 >= 5 are terms, being all solutions of the form (i,j,k) = (i, i-1, k) = (A001653(n+1), A182190(n), A001652(n)) for n >= 1.

%C There are other solutions too, like (i,j,k) = (104, 65, 55), (1364, 155, 154), (2295, 615, 594).

%e i = 5 is a term since 1/t_5 + 1/t_4 = 1/t_3, which is one of the A001653 cases.

%e i = 104 is a term since 1/t_104 + 1/t_65 = 1/t_55, and which is not from A001653.

%t tri[k_] := k*(k+1)/2; triQ[k_] := IntegerQ[k] && IntegerQ[Sqrt[8*k + 1]]; q[i_] := AnyTrue[Range[i-1], triQ[1/(1/tri[i] + 1/tri[#])] &]; Select[Range[1000], q] (* _Amiram Eldar_, Oct 25 2025 *)

%Y Cf. A000217, A001652, A001653, A182190.

%K nonn,more

%O 1,1

%A _Ctibor O. Zizka_, Oct 25 2025

%E a(10) from _Amiram Eldar_, Oct 25 2025