A053714 -id:A053714 - cp's OEIS frontend

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

Jeff Burch

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

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

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

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

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 *)

for(n=0,70, k=1; while(!isprime(n!+k), k++); print1(k,","))

a(n) = nextprime(n!+1) - n!; \\ Michel Marcus, Dec 25 2020

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

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

More terms from Jud McCranie
a(21) onwards from Wouter Meeussen
Better description from Rick L. Shepherd, Nov 06 2002

A033933 Least nonnegative m such that n! - m is prime.

Original entry on oeis.org

0, 1, 1, 7, 1, 1, 31, 13, 11, 13, 1, 23, 1, 47, 53, 59, 41, 101, 31, 31, 73, 89, 73, 149, 37, 43, 101, 31, 1, 61, 1, 1, 193, 113, 127, 97, 1, 73, 83, 131, 79, 109, 109, 53, 89, 79, 103, 59, 97, 179, 67, 59, 127, 61, 461, 277, 109, 137, 139, 71, 71, 101, 359, 127, 317, 191, 251, 103, 97, 751, 163, 373, 199, 167, 157, 491, 317

Offset: 2

Jeff Burch

Conjecture: for n >= 3, a(n) is 1 or a prime. - Amarnath Murthy, Mar 19 2002

a(n) is not divisible by any prime <= n. If a(n) > 1 is composite, then a(n) > n^2. There are no entries up to n = 2000 with a(n) > n^2, and there may be none. - Robert Israel, Jul 20 2014

Hans Havermann, Table of n, a(n) for n = 2..2000 (terms 2..500 from T. D. Noe)
Index entries for sequences related to factorial numbers

Cf. A002982, A006990, A033932, A056752, A053714.

0, seq(n! - prevprime(n!), n=3..100); # Robert Israel, Jul 15 2014

p[n_] := Module[{nf = n!}, nf - NextPrime[nf, -1]]; Join[{0}, Table[p[n], {n, 3, 70}]] (* Harvey P. Dale, Jul 07 2012 *)

for(n=2,70, k=0; while(!isprime(n!-k), k++); print1(k,","))

vector(66, t, my(n=t+1, f=n!); f-precprime(f)) \\ Joerg Arndt, Jul 19 2014

def A033933(n):
    if n < 3: return 0
    f = factorial(n)
    return f - previous_prime(f)
[A033933(n) for n in (2..78)] # Peter Luschny, Jul 20 2014

More terms from Jud McCranie
a(21) onwards from Wouter Meeussen
Corrected by Rick L. Shepherd, Nov 06 2002

A056752 Distance from n! to the nearest prime.

Original entry on oeis.org

1, 0, 1, 1, 7, 1, 1, 23, 13, 11, 1, 1, 23, 1, 43, 23, 31, 37, 89, 29, 31, 31, 89, 73, 41, 37, 1, 67, 31, 1, 61, 1, 1, 97, 61, 127, 1, 1, 73, 53, 1, 79, 71, 47, 53, 89, 79, 53, 59, 61, 179, 53, 59, 127, 61, 149, 107, 109, 137, 139, 71, 71, 101, 67, 127, 283, 73, 83, 103, 97

Offset: 1

Labos Elemer, Jan 19 2001

nonn

			For both 1! and 2! the nearest prime neighbor is 2, with distances of 1 and 0, respectively. The nearest primes around 8! are 40289 and 40343 with distances of 31 and 23, so a(8)=23.

Hans Havermann, Table of n, a(n) for n = 1..2000

Cf. A006990, A037151, A033932, A033933, A053714 (signed version with a different second term).

with(numtheory): [seq(min(nextprime(i!)-i!,i!-prevprime(i!)),i=3..100)]; # a(1) and a(2) computed individually

Table[Function[k, Min[k - #, NextPrime@ # - k] &@If[n == 1, 0, Prime@ PrimePi@ k]][n!], {n, 16}] (* Michael De Vlieger, Jul 15 2017 *)

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

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A033933 Least nonnegative m such that n! - m is prime.

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A056752 Distance from n! to the nearest prime.

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