A386369 a(1) = 0, and for any n > 1, a(n) is the largest k < n such that a(1) + ... + a(k) is a square number.

0, 1, 2, 2, 2, 2, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 18, 18, 18, 18, 18, 18, 18, 18, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26

Offset

1,3

Comments

In other words: a(1) = 0, and for any n > 0, if a(1) + ... + a(n) is a square number then a(n+1) = n, otherwise a(n+1) = a(n).

This sequence is unbounded: if a(1) + ... + a(n) = u^2, then a(n+1) = n, u^2 + n * (2*u+n) = (u+n)^2, so a(1) + ... + a(m) is a square number for some m > n, and a(m+1) = m > a(n+1).

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Example

Sequence begins:
  n   a(n)  a(1)+...+a(n)  Square?
  --  ----  -------------  -------
   1     0              0  Yes
   2     1              1  Yes
   3     2              3  No
   4     2              5  No
   5     2              7  No
   6     2              9  Yes
   7     6             15  No
   8     6             21  No
   9     6             27  No
  10     6             33  No

Mathematica

Module[{s = 0, a = 0}, Table[If[IntegerQ[Sqrt[s += a]], a = n-1]; a, {n, 100}]] (* Paolo Xausa, Jul 29 2025, after Rémy Sigrist *)

Prog

(PARI) { t = 0; v = 0; for (n = 1, 70, print1 (v", "); t += v; if (issquare(t), v = n; ); ); }

Crossrefs

See A385986 for a similar sequence.

Cf. A385254, A386370, A386687, A386688.

Sequence in context: A136494 A260188 A048764 * A327663 A248782 A038714

Adjacent sequences: A386366 A386367 A386368 * A386370 A386371 A386372

Keyword

nonn

Author

Rémy Sigrist, Jul 19 2025

Status

approved