A175194 -id:A175194 - 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

A092789 a(n) = smallest prime of the form prime(n)+m! for some m >= 0.

Original entry on oeis.org

3, 5, 7, 13, 13, 19, 19, 43, 29, 31, 37, 43, 43, 67, 53, 59, 61, 67, 73, 73, 79, 103, 89, 113, 103, 103, 109, 109, 229, 137, 151, 137, 139, 163, 151, 157, 163, 283, 173, 179, 181, 363061, 193, 199, 199, 223, 331, 229, 229, 349, 239, 241, 5281, 257, 263, 269, 271

Offset: 1

Cino Hilliard, Apr 14 2004

nonn

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

Is it known that such a prime always exists? If not the definition should say "or -1 if no such prime exists". - N. J. A. Sloane, Aug 11 2011

Cf. A175193, A175194, A082470.

SmallestP:=function(p) for m in [0..p-1] do q:=p+Factorial(m); if IsPrime(q) then return q; end if; end for; return -1; end function; [ SmallestP(NthPrime(n)): n in [1..80] ]; // Klaus Brockhaus, Mar 02 2010

A092789 := proc(n) local q,m ; for m from 0 do q := ithprime(n)+m! ; if isprime(q) then return q; end if; end do ; end proc:
seq(A092789(n),n=1..80) ; # R. J. Mathar, Mar 02 2010

nfactpm3(n) = { forprime(p=1,n, c=0; for(x=0,n,y=x!+p;if(isprime(y),c++;print1(y",");break)); ) }

Definition and offset corrected following a suggestion from Leroy Quet. - Klaus Brockhaus, Mar 02 2010

A175193 a(n) is the smallest positive integer such that (the n-th prime)+a(n)! is prime, or -1 if no such prime exists.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 2, 3, 3, 2, 4, 3, 3, 2, 3, 3, 2, 3, 4, 3, 4, 3, 2, 3, 2, 5, 4, 4, 3, 2, 4, 2, 3, 3, 5, 3, 3, 2, 9, 2, 3, 2, 4, 5, 3, 2, 5, 3, 2, 7, 3, 3, 3, 2, 3, 3, 2, 4, 4, 3, 2, 4, 8, 3, 5, 2, 4, 3, 4, 3, 3, 5, 3, 5, 4, 5, 4, 2, 5, 2, 3, 4, 3, 5, 3, 2, 4, 4, 4, 5, 7, 4, 3, 6, 2, 4, 3, 4, 3, 3, 2, 3

Offset: 1

Leroy Quet, Mar 01 2010

nonn

A175194(n) = a(n)!.

			From _Michael De Vlieger_, Nov 24 2017: (Start)
Records and their indices in a(n):
    i          n   a(n)
   --------------------
    1        1      1
    2        2      2
    3        4      3
    4        8      4
    5       29      5
    6       42      9
    7      233     10
    8      254     42
    9     4508     49
   10     7003    124
   11     7385    276
   12    60650    311
   13    97146    542
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, First positions of values and records in a(n).

Cf. A082470, A092789, A175194.

Fold[Append[#1, SelectFirst[Range[12], Function[k, PrimeQ[Prime[#2] + k!]]]] &, {1}, Range[2, 105]] (* Michael De Vlieger, Nov 24 2017 *)

a(n) = {my(k = 1, p = prime(n)); while (!isprime(p + k!), k++); k;} \\ Michel Marcus, Nov 25 2017

A092789(n) = A000040(n) + a(n)!.

Extended by Ray Chandler, Mar 04 2010

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

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A092789 a(n) = smallest prime of the form prime(n)+m! for some m >= 0.

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Maple

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A175193 a(n) is the smallest positive integer such that (the n-th prime)+a(n)! is prime, or -1 if no such prime exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions