A151800 Least prime > n (version 2 of the "next prime" function).

2, 2, 3, 5, 5, 7, 7, 11, 11, 11, 11, 13, 13, 17, 17, 17, 17, 19, 19, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 31, 31, 37, 37, 37, 37, 37, 37, 41, 41, 41, 41, 43, 43, 47, 47, 47, 47, 53, 53, 53, 53, 53, 53, 59, 59, 59, 59, 59, 59, 61, 61, 67, 67, 67, 67, 67, 67, 71, 71, 71, 71, 73, 73, 79

Offset: 0

Author

N. J. A. Sloane, Jun 29 2009

Keywords

• nonn
• easy
• nice

Comments

Version 1 of the "next prime" function is A007918: smallest prime >= n.

Maple's nextprime() is this version 2; PARI/GP's nextprime() is version 1.

See A007918 for references and further information.

a(n) is the smallest number greater than one that is not divisible by any 1 < k <= n. Consider a multi-round election in which, in each round, voters each cast one vote for one of the remaining candidates. Then, any candidates which receive the fewest votes in that round are eliminated. This repeats until either one candidate remains, who wins the election, or no candidates remain. a(n) is the smallest nontrivial number of voters that can guarantee a winner if the election initially has n > 0 candidates. This is a consequence of the first fact. - Thomas Anton, Mar 30 2020

Conjecture: if n > 3, then a(n) < n^(n^(1/n)). - Thomas Ordowski, Feb 23 2023

Links

• Daniel Forgues, Table of n, a(n) for n = 0..100000

Crossrefs

Cf. A000040, A007917, A007918, A061558, A066169, A151799, A317357, A036234.

Programs

Haskell

a151800 = a007918 . (+ 1)  -- Reinhard Zumkeller, Jul 26 2012

Magma

[NextPrime(n): n in [0..80]]; // Vincenzo Librandi, Jan 14 2016

Maple

map(nextprime,[$0..100]); # Robert Israel, Jul 15 2015

Mathematica

NextPrime[Range[0,80]] (* Harvey P. Dale, May 21 2011 *)

Maxima

makelist(next_prime(n), n, 0, 73); /* Bruno Berselli, May 20 2011 */

PARI

a(n)=nextprime(n+1) \\ Charles R Greathouse IV, Apr 28 2015

Python

from sympy import nextprime
def A151800(n):
    return nextprime(n) # Chai Wah Wu, Feb 28 2018

Formula

a(n) = A007918(n+1).

a(n) = 1 + Sum_{k=1..2n} (floor((n!^k)/k!) - floor(((n!^k)-1)/k!)). - Anthony Browne, May 11 2016

a(n) = A000040(A036234(n)). - Ridouane Oudra, Sep 30 2024