A007918 Least prime >= n (version 1 of the "next prime" function).

2, 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

Offset: 0

Author

R. Muller and Charles T. Le (charlestle(AT)yahoo.com)

Keywords

• nonn
• easy
• nice

Comments

Version 2 of the "next prime" function is "smallest prime > n". This produces A151800.

Maple uses version 2.

According to the "k-tuple" conjecture, a(n) is the initial term of the lexicographically earliest increasing arithmetic progression of n primes; the corresponding common differences are given by A061558. - David W. Wilson, Sep 22 2007

It is easy to show that the initial term of an increasing arithmetic progression of n primes cannot be smaller than a(n). - N. J. A. Sloane, Oct 18 2007

Also, smallest prime bounded by n and 2n inclusively (in accordance with Bertrand's theorem). Smallest prime >n is a(n+1) and is equivalent to smallest prime between n and 2n exclusively. - Lekraj Beedassy, Jan 01 2007

Run lengths of successive equal terms are given by A125266. - Felix Fröhlich, May 29 2022

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

Links

• T. D. Noe, Table of n, a(n) for n = 0..10000
• Jens Kruse Andersen, Records for primes in arithmetic progressions
• K. Atanassov, On Some of Smarandache's Problems
• K. Atanassov, On the 37th and 38th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 2, 83-85.
• Henry Bottomley, Prime number calculator
• J. Castillo, Other Smarandache Type Functions: Inferior/Superior Smarandache f-part of x, Smarandache Notions Journal, Vol. 10, No. 1-2-3, 1999, 202-204.
• Andrew Granville, Prime Number Patterns
• Hans Gunter, Puzzle 145. The Inferior Smarandache Prime Part and Superior Smarandache Prime Part functions; Solutions by Jean Marie Charrier, Teresinha DaCosta, Rene Blanch, Richard Kelley and Jim Howell.
• Jonathan Sondow and Eric Weisstein, Bertrand's Postulate, World of Mathematics.
• Eric Weisstein's World of Mathematics, Next Prime, k-tuple conjecture
• Index entries for sequences related to primes in arithmetic progressions

Crossrefs

Cf. A000040, A007917, A008407, A020497, A061558, A125266, A151799, A151800, A171400.

Programs

Haskell

a007918 n = a007918_list !! n
a007918_list = 2 : 2 : 2 : concat (zipWith
              (\p q -> (replicate (fromInteger(q - p)) q))
                                   a000040_list $ tail a000040_list)
-- Reinhard Zumkeller, Jul 26 2012

Magma

[2] cat [NextPrime(n-1): n in [1..80]]; // Vincenzo Librandi, Jan 14 2016

Maple

A007918 := n-> nextprime(n-1); # M. F. Hasler, Apr 09 2008

Mathematica

NextPrime[Range[-1, 72]] (* Jean-François Alcover, Apr 18 2011 *)

PARI

A007918(n)=nextprime(n)  \\ M. F. Hasler, Jun 24 2011

PARI

for(x=0,100,print1(nextprime(x)",")) \\ Cino Hilliard, Jan 15 2007

Python

from sympy import nextprime
def A007918(n): return nextprime(n-1) # Chai Wah Wu, Apr 22 2022

Formula

For n > 1: a(n) = A000040(A049084(A007917(n)) + 1 - A010051(n)). - Reinhard Zumkeller, Jul 26 2012

a(n) = A151800(n-1). - Seiichi Manyama, Apr 02 2018