A033932 -id:A033932 - cp's OEIS frontend

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

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

A087421 Smallest prime >= n!.

Original entry on oeis.org

2, 2, 2, 7, 29, 127, 727, 5051, 40343, 362897, 3628811, 39916801, 479001629, 6227020867, 87178291219, 1307674368043, 20922789888023, 355687428096031, 6402373705728037, 121645100408832089, 2432902008176640029, 51090942171709440031, 1124000727777607680031

Offset: 0

Mitch Cervinka (puritan(AT)planetkc.com), Oct 22 2003

nonn

n! is prime only when n=2. When n>2, for n!+m to be prime, m must be relatively prime to all the numbers from 2 to n. In particular, if m is between 2 and n, then (n!+m) will be divisible by m. Thus a(n) must be either n!+1, or else larger than n!+n.

			a(0) = 2 since 0! = 1 and 2 is the smallest prime >= 1.
a(4) = 29 since 4! = 24 and 29 is the smallest prime >= 24.

Hugo Pfoertner, Table of n, a(n) for n = 0..400
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.

Cf. A000142, A006990, A007918, A033932, A037151.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; Table[ NextPrim[n! - 1], {n, 0, 20}] (* Robert G. Wilson v, Oct 25 2003 *)
Join[{2,2,2},NextPrime[Range[3,25]!]]  (* Harvey P. Dale, Feb 23 2011 *)

a(n)=nextprime(n!); \\ R. J. Cano, Apr 08 2018

from sympy import factorial, nextprime
def a(n): return nextprime(factorial(n)-1)
print([a(n) for n in range(23)]) # Michael S. Branicky, May 22 2022

a(n) = min { p[i] | p[i]>=n! }, where p[i] is the set of prime numbers.

a(n) = A007918(A000142(n)). - Michel Marcus, Apr 09 2018

Edited, corrected and extended by Robert G. Wilson v and Ray Chandler, Oct 25 2003

A245695 Least number k >= 0 such that (n!+k)/n is prime.

Original entry on oeis.org

1, 2, 0, 4, 25, 42, 49, 88, 207, 170, 121, 12, 377, 938, 285, 688, 391, 558, 703, 1780, 609, 682, 713, 2328, 3275, 1066, 1593, 28, 1943, 6690, 3317, 4064, 2607, 1258, 3395, 2196, 4847, 38, 1677, 3880, 2173, 42, 4171, 3124, 2115, 10994, 4747, 11184, 2597, 4150, 3111, 14092, 2809, 3834, 12265, 3976, 8493, 6206, 16697, 17580, 16531, 47678, 8253, 17344, 4355, 12738, 18961, 4964, 5727, 9170, 9869, 61704, 7373

Offset: 1

Derek Orr, Jul 29 2014

nonn

a(n) = M*n for some integer M >= 0.

a(n) = n times least m >= 0 such that (n-1)!+m is prime. - Jens Kruse Andersen, Jul 30 2014

			(4!+0)/4 = 6 is not prime.
(4!+1)/4 = 25/4 is not prime.
(4!+2)/4 = 26/4 is not prime.
(4!+3)/4 = 27/4 is not prime.
(4!+4)/4 = 7 is prime. Thus a(4) = 4.

Jens Kruse Andersen, Table of n, a(n) for n = 1..2001

Cf. A033932, A245696, A245697.

a(n)=for(k=0,10^6,s=(n!+k)/n;if(floor(s)==s,if(ispseudoprime(s),return(k))))
n=1;while(n<100,print1(a(n),", ");n++)

a(n) = n*A033932(n-1), except a(3) = 0 where A033932 demands positive values. - Jens Kruse Andersen, Jul 30 2014

A339274 Number of times the n-th prime (=A000040(n)) occurs in A033933.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 0, 0, 1, 0, 4, 1, 1, 1, 1, 2, 3, 2, 1, 2, 3, 2, 2, 2, 3, 3, 4, 0, 4, 2, 4, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 0, 3, 3, 3, 2, 2, 1, 0, 4, 1, 2, 0, 2, 1, 3, 2, 4, 2, 2, 3, 4, 0, 4, 1, 3, 2, 2, 4, 0, 5, 2, 6, 2, 3, 3, 0, 5, 2, 4, 2, 3, 3, 1, 3, 2

Offset: 1

A.H.M. Smeets, Dec 25 2020

nonn

Each term in A033933 is either 1 or a prime number. Moreover it is known that each prime occurs only a finite number of times in A033933.

By excluding the terms that equal one from A033933, we observe the smallest value of A033933(n)/log(n!) in the range n = 3..2000 to be ~0.1552. From this it is believed that the primes less than 0.9*log(2001!)*0.1552 (~ 1846) will not occur anymore in the sequence A033933 for n > 2000; the applied factor 0.9 is a safety factor to be more or less sure that the prime numbers up to about 1846 will no longer occur in A033933.

			The prime number 13 occurs 2 times in A033933, and A000040(6) = 13, so a(6) = 2.

A.H.M. Smeets, Table of n, a(n) for n = 1..283
A.H.M. Smeets, Sum_{k = 1..n} a(k) versus A000040(n)

Cf. A000040, A033932, A033933, A339959.

A340013 The prime gap, divided by two, which surrounds n!.

Original entry on oeis.org

1, 3, 7, 4, 6, 27, 15, 11, 7, 15, 45, 10, 45, 38, 45, 39, 95, 30, 31, 52, 93, 102, 95, 48, 22, 84, 127, 54, 94, 40, 19, 145, 87, 129, 49, 22, 85, 68, 66, 88, 90, 78, 146, 95, 156, 78, 71, 79, 225, 60, 65, 175, 66, 305, 192, 196, 215, 205, 420, 101, 186, 213, 160

Offset: 3

Robert G. Wilson v, Jan 09 2021

nonn

A theorem states that between (n+1)! + 2 and (n+1)! + (n+1) inclusive, there are n consecutive composite integers, namely 2, 3, 4, ..., n, n+1.

Records: 1, 3, 7, 27, 45, 95, 102, 127, 145, 146, 156, 225, 305, 420, 804, 844, 1173, 1671, 1725, 1827, 2570, 2930, 3318, 5142, 5946, 6837, 7007, 8208, 10221, ..., .

			For a(1), there are no positive primes which surround 1!. Therefore a(1) is undefined.
For a(2), there are two contiguous primes {2, 3} with 2 being 2!. The prime gap is 1. However, the two primes do not surround 2!, so a(2) is undefined.
For a(3), the following set of numbers, {5, 6, 7}, with 3! being in the middle. The prime gap is 2; therefore, a(3) = 1.
For a(4), the following set of numbers, {23, 24, 25, 26, 27, 28, 29} with 4! in between the two primes 23 & 29. The prime gap is 6, so a(4) = 3.

Robert Israel, Table of n, a(n) for n = 3..607

Cf. A000142, A002981, A006990, A033932, A033933, A037151, A054588, A058054, A073308.

a:= n-> (f-> (nextprime(f-1)-prevprime(f+1))/2)(n!):
seq(a(n), n=3..70);  # Alois P. Heinz, Jan 09 2021

a[n_] := (NextPrime[n!, 1] - NextPrime[n!, -1])/2; Array[a, 70, 3]

a(n) = (nextprime(n!+1) - precprime(n!-1))/2; \\ Michel Marcus, Jan 11 2021

from sympy import factorial, nextprime, prevprime
def A340013(n):
    f = factorial(n)
    return (nextprime(f)-prevprime(f))//2 # Chai Wah Wu, Jan 23 2021

a(n) = (A037151(n) - A006990(n))/2 = (A033932(n) + A033933(n))/2.

a(n) = A054588(n)/2 = A058054(n)/2. - Alois P. Heinz, Jan 09 2021

A343717 a(n) is the smallest number that yields a prime when appended to n!.

Original entry on oeis.org

1, 1, 3, 1, 1, 1, 7, 11, 29, 17, 43, 29, 13, 47, 19, 73, 37, 19, 41, 103, 41, 31, 43, 1, 113, 31, 37, 59, 41, 53, 41, 47, 1, 41, 149, 37, 53, 73, 337, 1, 103, 151, 293, 47, 107, 509, 127, 71, 167, 197, 167, 149, 67, 163, 139, 251, 59, 107, 241, 331, 269, 1, 149

Offset: 0

Jon E. Schoenfield, May 17 2021

Appending to n! any number k <= n yields a multiple of k; that multiple cannot be prime except at k=1, so, for every n, a(n)=1 or a(n) > n.
a(n) = 1 iff n = 0 or n is in A024912.
See A068695 for a proof that a(n) always exists. - Felix Fröhlich, May 18 2021
If a(n) is composite, then a(n) > 2n. - Michael S. Branicky, May 18 2021

			n=1: 1! = 1; appending a 1 yields 11, a prime, so a(1)=1.
n=2: 2! = 2; appending a 1 yields 21 = 3*7, and appending a 2 yields 22 = 2*11, but appending a 3 yields 23 (a prime), so a(2)=3.
n=19: 19! = 121645100408832000; appending any number < 103 yields a composite, but 121645100408832000103 is a prime, so a(19)=103.

Lucas A. Brown, Table of n, a(n) for n = 0..2630 (terms 0..1000 from Michael S. Branicky).
Michael S. Branicky, Python program for b-file and A343718, A343719

Cf. A000040, A000142, A024912, A033932, A068695, A095194, A343718, A343719.

a:= proc(n) option remember; local k, t; t:= n!;
      for k while not isprime(parse(cat(t, k))) do od; k
    end:
seq(a(n), n=0..62);  # Alois P. Heinz, May 17 2021

Array[Block[{m = #!, k = 0}, While[! PrimeQ[10^If[k == 0, 1, IntegerLength[k]]*m + k], k++]; k] &, 62] (* Michael De Vlieger, May 17 2021 *)
snp[n_]:=Module[{nf=n!,c=1},While[!PrimeQ[nf*10^IntegerLength[c]+c],c++];c]; Array[snp,70,0] (* Harvey P. Dale, Oct 17 2024 *)

for(n=0,62,my(f=digits(n!));forstep(k=1,oo,2,my(p=fromdigits(concat(f,digits(k))));if(ispseudoprime(p),print1(k,", ");break))) \\ Hugo Pfoertner, May 18 2021

# see link for faster program producing b-file
from sympy import factorial, isprime
def a(n):
  start = str(factorial(n))
  end = 1
  while not isprime(int(start + str(end))): end += 2
  return end
print([a(n) for n in range(63)]) # Michael S. Branicky, May 17 2021

a(n) = A068695(n!) = A068695(A000142(n)).

A053712 Lower balancing primes to prime-balanced factorials.

Original entry on oeis.org

5, 113, 3628789, 51090942171709439969

Offset: 1

Labos Elemer, Feb 10 2000

nonn

The next two terms are 171!-397 and 190!-409. - Jud McCranie, Jul 04 2000

			113 is balancing 5! = 120 from below, where 5! = 120 is a balanced factorial.

Amiram Eldar, Table of n, a(n) for n = 1..7

Cf. A033932, A033933, A037151, A006990, A006562, A053709, A053710, A053711, A053713.

a(n) = A053709(n)! - A053711(n) = A053710(n) - A053711(n). - Amiram Eldar, Mar 10 2025

A053713 Upper balancing primes to prime-balanced factorials.

Original entry on oeis.org

7, 127, 3628811, 51090942171709440031

Offset: 1

Labos Elemer, Feb 10 2000

nonn

The next two terms are 171!+397 and 190!+409, which are too large to include. - Jud McCranie, Jul 04 2000

			127 is balancing 5! = 120 from above, where 5! = 120 is a balanced factorial.

Amiram Eldar, Table of n, a(n) for n = 1..7

Cf. A033932, A033933, A037151, A006990, A006562, A053709, A053710, A053711, A053712.

a(n) = A053709(n)! + A053711(n) = A053710(n) + A053711(n). - Amiram Eldar, Mar 10 2025

A053708 Nearest prime to n! (but not equal to n!).

Original entry on oeis.org

2, 3, 5, 23, 113, 719, 5039, 40343, 362867, 3628789, 39916801, 479001599, 6227020777, 87178291199, 1307674368043, 20922789888023, 355687428096031, 6402373705728037, 121645100408832089, 2432902008176640029, 51090942171709439969, 1124000727777607680031, 25852016738884976639911

Offset: 1

Labos Elemer, Feb 10 2000

nonn

If n! is the average of its closest prime neighbors then the smaller prime is to be chosen (as in A051701).

			For 8! = 40320 the closest upper and lower primes are 40289 and 40343 with d = 31 and d = 23, so 40343 is closer to 8! than the lower neighbor.

Amiram Eldar, Table of n, a(n) for n = 1..449

Cf. A000142, A033932, A037151, A033933, A006990.

f[n_]:=Module[{nf=n!,s,l},s=NextPrime[nf,-1];l=NextPrime[nf];If[nf-s>l-nf,l,s]]
Table[f[i],{i,25}] (* Harvey P. Dale, Dec 08 2010 *)

Corrected by Rick L. Shepherd, Jan 11 2006

a(21)-a(23) from Amiram Eldar, Mar 10 2025

A108518 a(n) is the smallest natural number m such that (10^n)! + m is prime.

Original entry on oeis.org

1, 11, 229, 1283, 44159

Offset: 0

Farideh Firoozbakht, Jul 10 2005

If a(n) is composite then a(n)>10^(2n)+2*10^n. Conjecture: All terms are noncomposite numbers.

(10^4)!+44159 is a probable prime. - Jason Yuen, May 20 2024

			a(3)=1283 because (10^3)!+1283 is prime and for 0<m<1283 1000!+m is
composite.

Cf. A108519, A033932.

a[n_] := (For[m = 1, ! PrimeQ[(10^n)! + m], m++ ]; m); Do[Print[a[n]], {n, 0, 3}]
sp[n_]:=Module[{c=(10^n)!},NextPrime[c]-c]; Array[sp,4,0] (* Harvey P. Dale, Jul 29 2013 *)

a(n) = A033932(10^n). - Jason Yuen, May 20 2024

a(4) from Jason Yuen, May 20 2024

cp's OEIS Frontend

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

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A087421 Smallest prime >= n!.

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A245695 Least number k >= 0 such that (n!+k)/n is prime.

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A339274 Number of times the n-th prime (=A000040(n)) occurs in A033933.

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A340013 The prime gap, divided by two, which surrounds n!.

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A343717 a(n) is the smallest number that yields a prime when appended to n!.

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A053712 Lower balancing primes to prime-balanced factorials.

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