A390113 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).

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

Offset

1,1

Comments

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.

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

Links

Table of n, a(n) for n=1..10.

Example

i = 5 is a term since 1/t_5 + 1/t_4 = 1/t_3, which is one of the A001653 cases.
i = 104 is a term since 1/t_104 + 1/t_65 = 1/t_55, and which is not from A001653.

Mathematica

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 *)

Crossrefs

Cf. A000217, A001652, A001653, A182190.

Sequence in context: A139856 A097345 A097344 * A153076 A034700 A085151

Adjacent sequences: A390110 A390111 A390112 * A390114 A390115 A390116

Keyword

nonn,more

Author

Ctibor O. Zizka, Oct 25 2025

Extensions

a(10) from Amiram Eldar, Oct 25 2025

Status

approved