A385986 a(1) = 2, and for any n > 1, a(n) is the largest k < n such that a(1) + ... + a(k) is prime.

2, 1, 2, 3, 3, 5, 5, 5, 5, 9, 9, 9, 9, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 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, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 65, 65, 65

Offset: 1

Author

Rémy Sigrist, Jul 14 2025

Keywords

• nonn
• easy

Comments

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

This sequence is unbounded: for any n > 1, let P = a(1) + ... + a(a(n)); P is prime and a(a(n)+1) = a(n); as P > a(n), P and a(n) are coprime, hence, by Dirichlet's theorem on arithmetic progressions, P + k*a(n) is prime for some minimal k > 0, and a(a(n)+k+1) = a(n)+k > a(n).

Examples

			Sequence begins:
  n   a(n)  a(1)+...+a(n)  Prime?
  --  ----  -------------  ------
   1     2              2  Yes
   2     1              3  Yes
   3     2              5  Yes
   4     3              8  No
   5     3             11  Yes
   6     5             16  No
   7     5             21  No
   8     5             26  No
   9     5             31  Yes
  10     9             40  No
  11     9             49  No
  12     9             58  No
  13     9             67  Yes
  14    13             80  No
		

Links

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

Crossrefs

See A385988 and A386369 for similar sequences.

Cf. A385987 (corresponding prime numbers).

Programs

Mathematica

v = 2;t = 0;values={};Do[AppendTo[values,v];t+=v;If[PrimeQ[t],v=n],{n, 1, 68}];values (* James C. McMahon, Jul 22 2025 *)

PARI

{ v = 2; t = 0; for (n = 1, 68, print1 (v", "); if (isprime(t += v), v = n);); }