A231152 Least integer k such that prime(n+1) + ... + prime(n+k) > prime(1) + ... + prime(n).

1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 25, 25, 26, 26, 26, 27, 27, 27, 28, 28, 29, 29, 29, 30, 30

Offset

1,2

Comments

a(n) is the least k such that 2*s(n) < s(n+k), where s(n) is the n-th partial sum of primes.

Links

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

Formula

a(n) = A231153(n) + 1.

Example

5 <= 2 + 3 < 5 + 7, so a(2) = 2.
19 + 23 <= 2 + 3 + 5 + 7 + 11 + 13 + 17 < 19 + 23 + 27, so a(7) = 3.

Maple

P:= [seq(ithprime(i), i=1..10^5)]:
S:= ListTools:-PartialSums(P):
f:= proc(n) local j;
    ListTools:-BinaryPlace(S, 2*S[n]+1/2)+1-n;
end proc:
map(f, [$1..200]); # Robert Israel, Mar 15 2026

Mathematica

test = 0; u = Table[test = test + Prime[n]; tmp = 0; NestWhile[# + 1 &, n, test >= (tmp = tmp + Prime[#+1]) &] - n, {n, 1, 80}]; u + 1  (* Peter J. C. Moses, Nov 02 2013 *)

Prog

(PARI) a(n) = my(s=vecsum(primes(n)), k=1); while (vecsum(primes(n+k)) - 2*s <= 0, k++); k; \\ Michel Marcus, Mar 18 2026

Crossrefs

Cf. A007504, A231151, A231153.

Sequence in context: A057810 A029917 A062300 * A086916 A008680 A120203

Adjacent sequences: A231149 A231150 A231151 * A231153 A231154 A231155

Keyword

nonn,easy

Author

Clark Kimberling, Nov 09 2013

Extensions

Corrected by Robert Israel, Mar 15 2026

Status

approved