A033932 Least positive m such that n! + m is prime.

1, 1, 1, 1, 5, 7, 7, 11, 23, 17, 11, 1, 29, 67, 19, 43, 23, 31, 37, 89, 29, 31, 31, 97, 131, 41, 59, 1, 67, 223, 107, 127, 79, 37, 97, 61, 131, 1, 43, 97, 53, 1, 97, 71, 47, 239, 101, 233, 53, 83, 61, 271, 53, 71, 223, 71, 149, 107, 283, 293, 271, 769, 131, 271

Offset

0,5

Comments

Conjecture: No term is a composite number. a(n) is a prime > 3*prime(k), where k is such that prime(k) < n <= prime(k+1). - Amarnath Murthy, Apr 07 2004

Terms after n = 2000 in the b-file correspond to Fermat and Lucas PRP. - Phillip Poplin, Oct 12 2019

Links

Phillip Poplin, Table of n, a(n) for n = 0..4000 (first 501 terms from T. D. Noe, then up to n=2000 from Hans Havermann)

Antonín Čejchan, Michal Křížek, and Lawrence Somer, On Remarkable Properties of Primes Near Factorials and Primorials, Journal of Integer Sequences, Vol. 25 (2022), Article 22.1.4.

Index entries for sequences related to factorial numbers

Formula

a(n) = A151800(n!) - n!. - Max Alekseyev, Jul 23 2014

Maple

a:= n-> (f-> nextprime(f)-f)(n!):
seq(a(n), n=0..70);  # Alois P. Heinz, Feb 22 2023

Mathematica

a[n_] := (an = 1; While[ !PrimeQ[n! + an], an++]; an); Table[a[n], {n, 0, 63}] (* Jean-François Alcover, Dec 05 2012 *)
NextPrime[#]-#&/@(Range[0, 70]!) (* Harvey P. Dale, May 18 2014 *)

Prog

(PARI) for(n=0, 70, k=1; while(!isprime(n!+k), k++); print1(k, ", "))
(PARI) a(n) = nextprime(n!+1) - n!; \\ Michel Marcus, Dec 25 2020
(Python)
from sympy import factorial, nextprime
def a(n): fn = factorial(n); return nextprime(fn) - fn
print([a(n) for n in range(64)]) # Michael S. Branicky, May 22 2022

Crossrefs

Cf. A000142, A002981, A033933, A037151, A037153, A056752, A053714, A151800.

Sequence in context: A159482 A231935 A216835 * A144186 A246458 A153979

Adjacent sequences: A033929 A033930 A033931 * A033933 A033934 A033935

Keyword

nice,nonn

Author

Jeff Burch

Extensions

More terms from Jud McCranie

a(21) onwards from Wouter Meeussen

Better description from Rick L. Shepherd, Nov 06 2002

Status

approved