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

Author

Rémy Sigrist, Jul 19 2025

Keywords

• nonn

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).

Examples

			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
		

Links

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

Crossrefs

See A385986 for a similar sequence.

Cf. A385254, A386370, A386687, A386688.

Programs

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 *)

PARI

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