A056752 -id:A056752 - 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

A053714 Smallest (in magnitude) nonzero number m such that n!+m is prime.

Original entry on oeis.org

1, 1, 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, -751, 101

Offset: 1

Labos Elemer, Feb 10 2000

sign

a(n) is the defined, nonzero (thus excluding a(1) and a(2) of A033933) minimum of A033932(n) and A033933(n) multiplied by -1 if that minimum is not A033932(n). If n!+m and n!-m are equidistant primes (A053709), we have (arbitrarily) chosen positive m.

			For n=4, the possible m are -1 (24-1) and +5 (24+5). The former is closer to 4! so a(4) is -1.
For n=5, the possible m are -7 (120-7) and +7 (120+7). Being equidistant to 5!, a(5) is +7.

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

Cf. A006990, A037151, A033932, A033933, A053709, A056752 (unsigned version with a different second term).

Edited by Hans Havermann, Jul 23 2014

A088412 A088258 indexed by A000142.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 11, 12, 14, 27, 30, 32, 33, 37, 38, 41, 73, 77, 94, 116, 154, 166, 320, 324, 340, 379, 399, 427, 469, 546, 872, 974, 1477, 1963, 3507, 3610, 6380, 6917, 21480, 26951, 34790, 94550, 103040, 110059, 147855, 150209, 208003

Offset: 1

Ray Chandler, Sep 29 2003

nonn

Union of A002981 and A002982, except 0. - Andrey Zabolotskiy, Aug 25 2016

Terms correspond to indices m where A056752(m)=1, excepting m=2. - Bill McEachen, May 20 2025

Hisanori Mishima, Primes near n!, Dec 2008.

Cf. A000142, A088258.

select(t -> isprime(t!-1) or isprime(t!+1), [$1..600]); # Robert Israel, Aug 25 2016

Select[Range[10^3], Or @@ PrimeQ@ {# - 1, # + 1} &[#!] &] (* Michael De Vlieger, Aug 25 2016 *)

a(n) is such positive k that A088258(n) = A000142(k).

Values 320 to 546 extracted from Mishima's table by R. J. Mathar, Mar 05 2010

More terms and correction of the initial term from Andrey Zabolotskiy, Aug 25 2016

cp's OEIS Frontend

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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A053714 Smallest (in magnitude) nonzero number m such that n!+m is prime.

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A088412 A088258 indexed by A000142.

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