A385254 Distinct terms in A386369.

0, 1, 2, 6, 18, 26, 70, 82, 222, 642, 1902, 5682, 17022, 51042, 59778, 59958, 77774, 107258, 268870, 285010, 361086, 930666, 1084314, 1498134, 3813282, 5053994, 13240150, 14183598, 15487758, 15579122, 18418666, 18622506, 23809998, 58728474, 64572254, 65013058

Offset: 1

Author

David A. Corneth and Paolo Xausa, Jul 29 2025

Keywords

• nonn

Comments

a(1) = 0. a(n) is the smallest positive integer > a(n-1) such that Sum_{m = 1..n-1} (a(m+1)-a(m))*a(m) is a perfect square.

Examples

			The first 6 terms of A386369 are 0, 1, 2, 2, 2, 2 which has partial sum 9. We have A386369(7) = 6. To find a(4) we look for the next term in A386369 that is larger than 6 i.e. solve 6*(k-6) + 9 = s^2 for some k. Rewrite gives 6*(k-6) = s^2 - 9 = (s-3)(s + 3). So we have 4 cases:
   1 | s - 3, 6 | s + 3
   2 | s - 3, 3 | s + 3
   3 | s - 3, 2 | s + 3
   6 | s - 3, 1 | s + 3
Solving for smallest t > 6 gives s = 9. So 6*(k-6) = 9^2 - 9 = 72 and so k = 18.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..1018
• David A. Corneth, PARI program

Crossrefs

Cf. A386369.

Programs

Mathematica

Module[{s = 0, a = 0}, Table[If[IntegerQ[Sqrt[s += a]], a = k-1, Nothing], {k, 10^5}]]

PARI

\\ See Corneth link

Extensions

More terms from Michael De Vlieger, Jul 29 2025