A007917 Version 1 of the "previous prime" function: largest prime <= n.

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

Offset

2,1

Comments

Version 2 of the "previous prime" function (see A151799) is "largest prime < n". This produces the same sequence of numerical values, except the offset (or indexing) starts at 3 instead of 2.

Maple's "prevprime" function uses version 2.

Also the largest prime dividing n! or lcm(1,...,n). - Labos Elemer, Jun 22 2000

Also largest prime among terms of (n+1)st row of Pascal's triangle. - Jud McCranie, Jan 17 2000

Also largest integer k such that A000203(k) <= n+1. - Benoit Cloitre, Mar 17 2002. - Corrected by Antti Karttunen, Nov 07 2017

Also largest prime factor of A061355(n) (denominator of Sum_{k=0..n} 1/k!). - Jonathan Sondow, Jan 09 2005

Also prime(pi(x)) where pi(x) is the prime counting function = number of primes <= x. - Cino Hilliard, May 03 2005

Also largest prime factor, occurring to the power p, in denominator of Sum_{k=1..n} 1/k^p, for any positive integer p. - M. F. Hasler, Nov 10 2006

For n > 10, these values are close to the most negative eigenvalues of A191898 (conjecture). - Mats Granvik, Nov 04 2011

References

K. Atanassov, On the 37th and the 38th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 2, 83-85.

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.

Links

N. J. A. Sloane, Table of n, a(n) for n = 2..10000

Marc Deleglise and Jean-Louis Nicolas, Maximal product of primes whose sum is bounded, arXiv preprint arXiv:1207.0603 [math.NT], 2012. See Fig. 1. - From N. J. A. Sloane, Dec 17 2012

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. F. Smarandache, Only Problems, Not Solutions!.

Mihai Prunescu and Joseph M. Shunia, Elementary closed-forms for non-trivial divisors, arXiv:2510.26939 [math.NT], 2025. See p. 8.

Eric Weisstein's World of Mathematics, Previous Prime

Formula

Equals A006530(A000142(n)). - Jonathan Sondow, Jan 09 2005

Equals A006530(A056040(n)). - Peter Luschny, Mar 04 2011

a(n) = A000040(A049084(A007918(n)) + 1 - A010051(n)). - Reinhard Zumkeller, Jul 26 2012

From Wesley Ivan Hurt, May 22 2013: (Start)

omega( Product_{i=2..n} a(i) ) = pi(n).

Omega( Product_{i=2..n} a(i) ) = n - 1. (End)

For n >= 2, a(A000203(n)) = A070801(n). - Antti Karttunen, Nov 07 2017

a(n) = n + 1 - Sum_{i=1..n} floor(pi(i)/pi(n)) = n + 1 - A175851(n). - Ridouane Oudra, Jun 24 2024

Conjecture: a(n) = floor(log(Sum_{k=2..n} exp(A000010(k)+1))). - Joseph M. Shunia, Aug 09 2024

a(n) = A000040(A000720(n)). - Ridouane Oudra, Oct 04 2024

Maple

A007917 := n-> prevprime(n+1);

Mathematica

Table[Prime[PrimePi[n]], {n, 2, 70}] (* Stefan Steinerberger, Apr 06 2006 *)
NextPrime[Range[3, 80], -1] (* Harvey P. Dale, Jan 23 2011 *)

Prog

(PARI) a=precprime \\ In older versions, use a(n)=precprime(n)
\\ Charles R Greathouse IV, Jun 15 2011
(Haskell)
a007917 n = if a010051' n == 1 then n else a007917 (n-1)
-- Reinhard Zumkeller, May 01 2013, Jul 26 2012
(Magma) [NthPrime(#PrimesUpTo(n)): n in [2..100]]; // Vincenzo Librandi, Nov 25 2015

Crossrefs

Cf. A000040, A000203, A005145, A007918, A070801, A113523, A151799, A179278, A175851, A000720.

Sequence in context: A385925 A394354 A136548 * A151799 A305429 A093841

Adjacent sequences: A007914 A007915 A007916 * A007918 A007919 A007920

Keyword

nonn,easy,nice

Author

R. Muller

Extensions

Edited by N. J. A. Sloane, Apr 06 2008

Status

approved