A121861 Least previously nonoccurring positive integer such that partial sum + 1 is prime.

1, 3, 2, 4, 6, 12, 8, 10, 14, 18, 22, 26, 24, 16, 30, 32, 28, 20, 34, 36, 42, 44, 46, 62, 52, 38, 60, 48, 58, 56, 54, 40, 50, 64, 68, 72, 76, 84, 66, 96, 74, 70, 80, 100, 86, 78, 88, 104, 90, 106, 122, 112, 98, 102, 94, 92, 118, 114, 108, 110, 124, 116, 138, 82, 120, 128, 150

Offset: 1

Author

Jonathan Vos Post, Aug 30 2006

Keywords

• easy
• nonn

Comments

Conjecture: a(n) = {1,3} UNION {permutation of even positive numbers}.

The corresponding partial sums + 1 are 2, 5, 7, 13, 17, 29, 37, 47, 61, 79, 101, 127, 151, ...,.

Examples

			a(1) = 1 because 1+1 = 2 is prime.
a(2) = 3 because 1+3+1 = 5 is prime.
a(3) = 2 because 1+3+2+1 = 7 is prime.
a(4) = 4 because 1+3+2+4+1 = 11 is prime.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A121862.

Programs

Maple

N:= 200: # to get all terms before the first term > N
A[1]:= 1: A[2]:= 3: P:= 5; S:= [seq(2*i,i=1..N/2)]:
for n from 3 while assigned(A[n-1]) do
  for k from 1 to nops(S) do
    if isprime(P+S[k]) then
      A[n]:= S[k];
      P:= P + S[k];
      S:= subsop(k=NULL,S);
      break
    fi
  od;
od:
seq(A[i],i=1..n-2); # Robert Israel, May 02 2017

Mathematica

f[s_] := Append[s, k = 1; p = 1 + Plus @@ s; While[MemberQ[s, k] || ! PrimeQ[p + k], k++ ]; k]; Nest[f, {}, 67] (* Robert G. Wilson v *)

Formula

a(n) = MIN{k>0 such that 1 + k + SUM[i=1..n-1]a(i) is prime and k <> a(i)}.

Extensions

Corrected and extended by Robert G. Wilson v, Aug 31 2006

Comment edited by Robert Israel, May 02 2017