A352912 -id:A352912 - cp's OEIS frontend

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

Jeff Burch, Apr 27 2003

nonn

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

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

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

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

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) ;

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

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

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

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

Offset corrected by Robert Israel, May 26 2021

A352913 a(n) = largest prime of the form prime(n) + k! (k >= 0).

Original entry on oeis.org

3, 5, 29, 727, 3628811, 733, 39916817, 87178291219, 20922789888023, 2432902008176640029, 1124000727777607680031, 8683317618811886495518194401280000037, 15511210043330985984000041, 523022617466601111760007224100074291200000043, 2658271574788448768043625811014615890319638528000000047

Offset: 1

Editors of OEIS, based on a suggestion from Hemjyoti Nath, Apr 16 2022

nonn

Michael S. Branicky, Table of n, a(n) for n = 1..93

These are the final entries in the rows of the triangle in A352912. See also A082470.

from sympy import isprime, prime
from itertools import count, islice
def agen(): # generator of terms
    for n in count(1):
        pn, fk = prime(n), 1
        for k in range(1, pn+1):
            if isprime(pn + fk): yield pn + fk
            fk *= k
print(list(islice(agen(), 51))) # Michael S. Branicky, Apr 16 2022

A384181 Primes p such that k! + p or |k! - p| is composite for all k >= 0.

Original entry on oeis.org

2, 3, 71, 97, 179, 181, 211, 223, 251, 283, 431, 503, 577, 827, 857, 971, 1019, 1021, 1109, 1213, 1249, 1259, 1279, 1289, 1373, 1427, 1429, 1483, 1571, 1609, 1619, 1637, 1699, 1709, 1759, 1801, 2053, 2129, 2141, 2213, 2269, 2281, 2293, 2297, 2339, 2381, 2477, 2503

Offset: 1

Gonzalo Martínez, May 21 2025

nonn

It is unknown whether there exists a prime p such that k! + p is composite for all k > = 0 (see A082470).

Every prime p in this list satisfies that at least one of the numbers k! + p, |k! - p| is composite; i.e., they cannot both be prime, for k >= 0.

			71 is in this sequence, since k! + 71 is prime only when k = 2, 5, 9, 14, 22, 43, 53 and 55, but |k! - 71| is composite for such values of k.

Cf. A082470, A352912.

from sympy import isprime, primerange, factorial
def ok(p):
    return not any(isprime((fk := factorial(k)) + p) and isprime(abs(fk - p)) for k in range(1, p))
print([p for p in primerange(2, 500) if ok(p)])

a(17)-a(23) from Sean A. Irvine, May 28 2025

a(24)-a(48) from Michael S. Branicky, May 29 2025

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

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A352913 a(n) = largest prime of the form prime(n) + k! (k >= 0).

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A384181 Primes p such that k! + p or |k! - p| is composite for all k >= 0.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Python

Extensions