A102781 Number of positive even numbers less than the n-th prime.

0, 1, 2, 3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36, 39, 41, 44, 48, 50, 51, 53, 54, 56, 63, 65, 68, 69, 74, 75, 78, 81, 83, 86, 89, 90, 95, 96, 98, 99, 105, 111, 113, 114, 116, 119, 120, 125, 128, 131, 134, 135, 138, 140, 141, 146, 153, 155, 156

Offset: 1

Author

Cino Hilliard, Feb 25 2005

Keywords

• easy
• nonn

Comments

Same as A005097 ((odd primes - 1)/2) with a leading zero. - Lambert Klasen, Nov 06 2005

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005097. - R. J. Mathar, May 18 2009

Equals (A000040-1)/2, integer part (0) for the first term. - M. F. Hasler, Dec 13 2019

Programs

Mathematica

Table[Prime[n] - Floor[Prime[n]/2] - 1, {n, 65}] (* Robert G. Wilson v *)

PARI

c(n,r) = { local(p); forprime(p=r,n, print1(floor(primorial(p)/ primorial(p-r)/primorial(r)+.0)",") ) } primorial(n) = \ The product primes <= n using the pari primelimit. { local(p1,x); if(n==0||n==1, return(2)); p1=1; forprime(x=2,n,p1*=x); return(p1) }

PARI

apply( A102781(n)=(prime(n)-1)\2, [1..99]) \\ M. F. Hasler, Dec 13 2019

Python

from sympy import prime
def A102781(n): return prime(n)-1>>1 # Chai Wah Wu, Oct 13 2024

Formula

Integer part of p#/((p-2)#*2#), where p=prime(n) and i# is the primorial function A034386(i). - Cino Hilliard, Feb 25 2005

n# = product of primes <= n. 0# = 1# = 2. [This is not a standard convention!] n#/(n-r)#/r# is analogous to the number of binomial coefficients A007318 = C(n, r) = n!/(n-r)!/r! where factorial ! is replaced by primorial #.

a(n) = prime_n - floor((prime_n)/2) - 1. - Giovanni Teofilatto, Nov 05 2005

a(n) = [A034386(prime(n))/(2*A034386(prime(n)-2))], n>2. - R. J. Mathar, May 18 2009

a(n) = [(prime(n)-1)/2] where the integer part [.] needs be taken only for n=1. - M. F. Hasler, Dec 13 2019

Extensions

Simpler definition from Giovanni Teofilatto, Nov 05 2005

Edited by N. J. A. Sloane Jul 05 2009 at the suggestion of R. J. Mathar