A386905 -id:A386905 - cp's OEIS frontend

A385988 a(1) = 1, and for any n > 1, a(n) is the largest k < n such that a(1) + ... + a(k) is a power of 2.

1, 1, 2, 3, 3, 3, 3, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23

Offset: 1

Rémy Sigrist, Jul 14 2025

nonn

In other words: a(1) = 1, and for any n > 0, if a(1) + ... + a(n) is a power of 2 then a(n+1) = n, otherwise a(n+1) = a(n).

This sequence is unbounded: if a(1) + ... + a(n) = 2^i, let n = m * 2^j with m odd, note that n <= 2^i, so j <= i, let z be the multiplicative order of m modulo 2, let t = 2^(i-j) * (2^z - 1) / m, a(1) + ... + a(n) + t * n = 2^(z+i), so a(1) + ... + a(n) + u * n is a power of 2 for some u in the range n+1..t, and a(u+1) > a(n+1) = n.

			Sequence begins:
  n   a(n)  a(1)+...+a(n)  Power of 2?
  --  ----  -------------  -----------
   1     1              1  Yes
   2     1              2  Yes
   3     2              4  Yes
   4     3              7  No
   5     3             10  No
   6     3             13  No
   7     3             16  Yes
   8     7             23  No
   9     7             30  No
  10     7             37  No
  11     7             44  No
  12     7             51  No

Paolo Xausa, Table of n, a(n) for n = 1..10000

See A385986 for a similar sequence.

Cf. A386905 (distinct terms).

Module[{s = -1, a = 1}, Table[If[DigitSum[s += a, 2] == 1, a = n - 1]; a, {n, 100}]] (* Paolo Xausa, Aug 07 2025 *)

{ v = 1; t = 0; for (n = 1, 71, print1(v", "); if (hammingweight(t += v)==1, v = n;);); }

cp's OEIS Frontend

A385988 a(1) = 1, and for any n > 1, a(n) is the largest k < n such that a(1) + ... + a(k) is a power of 2.

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