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

1,3

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

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

Mathematica

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

Prog

(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

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

Sequence in context: A082583 A274332 A005097 * A130290 A139791 A027563

Adjacent sequences: A102778 A102779 A102780 * A102782 A102783 A102784

Keyword

easy,nonn

Author

Cino Hilliard, Feb 25 2005

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

Status

approved