A006990 -id:A006990 - cp's OEIS frontend

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

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

A000197 a(n) = (n!)!.

Original entry on oeis.org

1, 1, 2, 720, 620448401733239439360000

Offset: 0

N. J. A. Sloane

The sequence 1, 2, 720!, 4!!!!, ... ,n!!...! (n times) grows too rapidly to have its own entry. See Hofstadter.
a(n) is divisible by 2^A245087(n) but not by 2^(A245087(n)+1), A245087 being the number of trailing zeros in its binary expansion. Also, for n>1, the largest prime divisor of a(n) is the largest prime <= n!, which is listed in A006990(n). - Stanislav Sykora, Jul 14 2014
See b-file for a(5), which has 199 digits and is too large to include. - Jianing Song, Jun 28 2018

Archimedeans Problems Drive, Eureka, 37 (1974), 11.
Douglas R. Hofstadter, Fluid concepts & creative analogies: computer models of the fundamental mechanisms of thought, Basic Books, 1995, pages 44-46. [From Colin Rowat, Sep 30 2011]
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Franklin T. Adams-Watters, Table of n, a(n) for n=0..5
Rudolph Ondrejka, 1273 exact factorials, Math. Comp., 24 (1970), 231.
Eric Weisstein's World of Mathematics, Factorial.
Index entries for sequences related to factorial numbers

Cf. A063979. - Robert G. Wilson v, Dec 04 2008
Cf. A152168. - Alois P. Heinz, Aug 04 2013
Cf. A000142, A006990, A245087.

[Factorial(Factorial(n)) : n in [0..5]]; // Wesley Ivan Hurt, Jul 14 2014

A000197:=n->(n!)!: seq(A000197(n), n=0..5); # Wesley Ivan Hurt, Jul 14 2014

Table[(n!)!, {n, 0, 5}] (* Wesley Ivan Hurt, Jul 14 2014 *)

a(n) = A000142(A000142(n)). - Wesley Ivan Hurt, Jul 14 2014

Sum_{n>=0} 1/a(n) = A336686. - Amiram Eldar, Mar 10 2021

A003604 Number of primes <= n!.

Original entry on oeis.org

0, 0, 1, 3, 9, 30, 128, 675, 4231, 30969, 258689, 2428956, 25306287, 289620751, 3610490805, 48686912930, 706003798139, 10953617995740, 181035032207760, 3175094503778521, 58893601709552538, 1151825702178908788, 23688535118132456668, 511050155316058710033

Offset: 0

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

Number of distinct prime divisors of (n!)!, (A000197). - Jason Earls, Jul 04 2001

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David Baugh, Table of n, a(n) for n = 0..25 (a(24)-a(25) found using Kim Walisch's primecount program).
Andrew R. Booker, The Nth Prime Page
Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x)
R. G. Wilson v, Letter to N. J. A. Sloane, Oct. 1993
R. G. Wilson, V, Letter to N. J. A. Sloane, Oct. 1993
R. G. Wilson v, Letter to N. J. A. Sloane, Jan. 1994
Index entries for sequences related to numbers of primes in various ranges

Cf. A000040, A000197.

Mathematica
```
Table[PrimePi[n!], {n, 0, 16}]
```
PARI
```
for(n=0,10,print(omega(n!!)))
```

a(n)=primepi(n!) \\ Charles R Greathouse IV, Jan 21 2016

[prime_pi(factorial(n)) for n in range(0, 14)] # Zerinvary Lajos, Jun 06 2009

a(15) from Jud McCranie
a(16)-a(17) from Paul Zimmermann
a(18) from Donovan Johnson, Dec 18 2009
a(19) from Donovan Johnson, Feb 18 2010
a(20) from Henri Lifchitz, Nov 11 2012
a(21)-a(23) from Henri Lifchitz, Aug 26 2014

A053709 Prime balanced factorials: numbers k such that k! is the mean of its 2 closest primes.

Original entry on oeis.org

3, 5, 10, 21, 171, 190, 348, 1638, 3329

Offset: 1

Labos Elemer, Feb 10 2000

Also, the integers k such that A033932(k) = A033933(k).
k! is an interprime, i.e., the average of two successive primes.
The difference between k! and any of its two closest primes must be 1 or exceed k. - Franklin T. Adams-Watters
Larger terms may involve probable primes. - Hans Havermann, Aug 14 2014

			For the 1st term, 3! is in the middle between its closest prime neighbors 5 and 7.
For the 2nd term, 5! is in the middle between its closest prime neighbors 113 and 127.
From _Jon E. Schoenfield_, Jan 14 2022: (Start)
In the table below, k = a(n), k! - d and k! + d are the two closest primes to k!, and d = A033932(k) = A033933(k) = A053711(n):
.
  n     k     d
  -  ----  ----
  1     3     1
  2     5     7
  3    10    11
  4    21    31
  5   171   397
  6   190   409
  7   348  1657
  8  1638  2131
  9  3329  7607
(End)

Cf. A053711 (distances), A033932, A033933, A006990, A037151, A006562, A053710, A075275.

Cf. A075409 (smallest m such that n!-m and n!+m are both primes).

for n from 3 to 200 do j := n!-prevprime(n!): if not isprime(n!+j) then next fi: i := 1: while not isprime(n!+i) and (i<=j) do i := i+2 od: if i=j then print(n):fi:od:

PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ[k], k-- ]; k]; NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k] Do[ a = n!; If[2a == PrevPrim[a] + NextPrim[a], Print[n]], {n, 3, 415}]

a(5)-a(6) from Jud McCranie, Jul 04 2000
a(7) from Robert G. Wilson v, Sep 17 2002
a(8) from Donovan Johnson, Mar 23 2008
a(9) from Hans Havermann, Aug 14 2014

A053710 Prime-balanced factorials: factorials k! that are the mean of their 2 closest neighboring primes.

Original entry on oeis.org

6, 120, 3628800, 51090942171709440000

Offset: 1

Labos Elemer, Feb 10 2000

nonn

Values of k are in A053709.

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

			For k = 21, k! = 51090942171709440000, d = 31, and the closest primes to 21! are q = 21! - 31 = 51090942171709439969, p = 21! + 31 = 51090942171709440031.

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

Cf. A033932, A033933, A006990, A037151, A006562, A053709.

Select[Range[25]!,NextPrime[#]-#==#-NextPrime[#,-1]&] (* Harvey P. Dale, May 08 2025 *)

k! = (p+q)/2; p = k! + d, q = k! - d, where p and q are the closest primes to k!.

a(n) = A053709(n)!.

a(3) corrected by Sean A. Irvine, Jan 14 2022

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 *)

A058054 Smallest prime > n! minus largest prime <= n!.

Original entry on oeis.org

1, 2, 6, 14, 8, 12, 54, 30, 22, 14, 30, 90, 20, 90, 76, 90, 78, 190, 60, 62, 104, 186, 204, 190, 96, 44, 168, 254, 108, 188, 80, 38, 290, 174, 258, 98, 44, 170, 136, 132, 176, 180, 156, 292, 190, 312, 156, 142, 158, 450, 120, 130, 350, 132, 610, 384, 392, 430

Offset: 2

Labos Elemer, Nov 20 2000

nonn

			For n = 2, 3, 4, 5, A037151(n) = 3, 7, 29, 127 and A006990(n) = 2, 5, 23, 113. The differences are: 1, 2, 6, 14.

Hans Havermann, Table of n, a(n) for n = 2..2000

Cf. A006990, A037151, A033932, A033933.

Essentially the same as A054588.

[seq(nextprime(i!)-prevprime(i!+1), i=2...100)];

f[n_] := NextPrime[n!] - NextPrime[n!, -1]; Array[f, 70, 3] (* Robert G. Wilson v, Jul 23 2014 *)

a(n) = A037151(n) - A006990(n)

a(n) = A033932(n) + A033933(n)

Edited by Hans Havermann, Jul 23 2014

A053711 Numbers d such that, for some k, the upper and lower primes closest to k! are k! + d and k! - d.

Original entry on oeis.org

1, 7, 11, 31, 397, 409, 1657, 2131, 7607

Offset: 1

Labos Elemer, Feb 10 2000

This sequence lists d = nextprime(k!) - k! = prevprime(k!) - k! for k in A053709.

			For k = 10, k! = 3628800, d = 11, and the closest primes to 10! are q = 10! - 11 = 3628789 and p = 10! + 11 = 3628811. The differences d are listed here.

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

Reap[Do[If[SameQ @@ #, Sow@ First[#]] &@ Abs[# - NextPrime[#, {-1, 1}]] &[i!], {i, 200}]][[-1, -1]] (* Michael De Vlieger, Jan 14 2022 *)

a(5)-a(8) from Donovan Johnson, Oct 12 2008

a(9) from Hans Havermann, Aug 15 2014

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

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

cp's OEIS Frontend

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

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A000197 a(n) = (n!)!.

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A003604 Number of primes <= n!.

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A053709 Prime balanced factorials: numbers k such that k! is the mean of its 2 closest primes.

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A053710 Prime-balanced factorials: factorials k! that are the mean of their 2 closest neighboring primes.

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A056752 Distance from n! to the nearest prime.

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A058054 Smallest prime > n! minus largest prime <= n!.

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