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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.

%I A386369 #13 Jul 30 2025 09:47:52
%S A386369 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,
%T A386369 26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,
%U A386369 26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26
%N 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.
%C A386369 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).
%C A386369 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).
%H A386369 Rémy Sigrist, <a href="/A386369/b386369.txt">Table of n, a(n) for n = 1..10000</a>
%e A386369 Sequence begins:
%e A386369   n   a(n)  a(1)+...+a(n)  Square?
%e A386369   --  ----  -------------  -------
%e A386369    1     0              0  Yes
%e A386369    2     1              1  Yes
%e A386369    3     2              3  No
%e A386369    4     2              5  No
%e A386369    5     2              7  No
%e A386369    6     2              9  Yes
%e A386369    7     6             15  No
%e A386369    8     6             21  No
%e A386369    9     6             27  No
%e A386369   10     6             33  No
%t A386369 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_ *)
%o A386369 (PARI) { t = 0; v = 0; for (n = 1, 70, print1 (v", "); t += v; if (issquare(t), v = n;);); }
%Y A386369 See A385986 for a similar sequence.
%Y A386369 Cf. A385254, A386370, A386687, A386688.
%K A386369 nonn
%O A386369 1,3
%A A386369 _Rémy Sigrist_, Jul 19 2025