A385986 -id:A385986 - cp's OEIS frontend

A385987 Prime numbers of the form A385986(1) + ... + A385986(k) for some k > 0.

2, 3, 5, 11, 31, 67, 197, 887, 1523, 1783, 2473, 2789, 3121, 3469, 3833, 4783, 4993, 5849, 6079, 6547, 7757, 8543, 9091, 10501, 12011, 16519, 18787, 19993, 20407, 21661, 22091, 23827, 27427, 29837, 30841, 32371, 33937, 34471, 37699, 38261, 42223, 43411, 50033

Offset: 1

Rémy Sigrist, Jul 14 2025

nonn

			A385986(1) + ... + A385986(5) = 2 + 1 + 2 + 3 + 3 = 11 is prime, so 11 appears in this sequence.

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

Cf. A385986.

{ v = 2; t = 0; for (n = 1, 323, if (isprime(t += v), print1 (t", "); v = 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.

Original entry on oeis.org

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

Rémy Sigrist, Jul 19 2025

nonn

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).

			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

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

See A385986 for a similar sequence.

Cf. A385254, A386370, A386687, A386688.

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 *)

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

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.

Original entry on oeis.org

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

A385987 Prime numbers of the form A385986(1) + ... + A385986(k) for some k > 0.

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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.

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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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