A082470 a(n) is the number of k >= 0 such that k! + prime(n) is prime.

2, 1, 3, 4, 5, 3, 6, 7, 6, 6, 9, 11, 9, 5, 10, 9, 10, 9, 9, 8, 9, 9, 11, 8, 10, 10, 12, 16, 12, 10, 10, 13, 14, 14, 16, 11, 12, 9, 15, 10, 9, 8, 12, 9, 10, 6, 8, 7, 14, 13, 10, 21, 15, 9, 13, 11, 9, 19, 12, 13, 16, 11, 19, 17, 9, 13

Offset: 1

Author

Jeff Burch, Apr 27 2003

Keywords

• nonn

Comments

k! + p is composite for k >= p since p divides k! for k >= p.

The first 10^6 terms are nonzero. Remarkably, the number 7426189 + m! is composite for all m <= 1793. - T. D. Noe, Mar 02 2010

Apparently it is not known whether a(n) is ever zero. - N. J. A. Sloane, Aug 11 2011

Examples

			For n = 4, 3!+7 = 13, 4!+7=31, 5!+7=127 and 6!+7 = 727 are the 4 primes in n!+7.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..227

Crossrefs

Cf. A092789, A175193, A175194, row lengths of A352912.

Programs

Maple

A082470 := proc(n)
    local ctr,j ;
    ctr := 0:
    for j from 0 to ithprime(n)-1 do
        if isprime(j!+ithprime(n))=true then
            ctr := ctr+1
        end if ;
    end do ;
    ctr
end proc:
seq(A082470(n),n=1..50) ;

Mathematica

Table[Count[Range[0,Prime[n]-1]!+Prime[n],?PrimeQ],{n,70}] (* _Harvey P. Dale, Feb 06 2019 *)

PARI

nfactppct(n) = { forprime(p=1,n, c=0; for(x=0,n,y=x!+p;if(isprime(y),c++) ); print1(c",") ) } \\ Cino Hilliard, Apr 15 2004

Python

from sympy import isprime, prime
from itertools import count, islice
def agen(): # generator of terms
    for n in count(1):
        pn, fk, an = prime(n), 1, 0
        for k in range(1, pn+1):
            if isprime(pn + fk): an += 1
            fk *= k
        yield an
print(list(islice(agen(), 40))) # Michael S. Branicky, Apr 16 2022

Extensions

Edited by Franklin T. Adams-Watters, Aug 01 2006

Offset corrected by Robert Israel, May 26 2021