A033933 -id:A033933 - cp's OEIS frontend

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

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.

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

Original entry on oeis.org

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

A006990 Largest prime <= n!.

Original entry on oeis.org

2, 5, 23, 113, 719, 5039, 40289, 362867, 3628789, 39916787, 479001599, 6227020777, 87178291199, 1307674367953, 20922789887947, 355687428095941, 6402373705727959, 121645100408831899, 2432902008176639969, 51090942171709439969, 1124000727777607679927

Offset: 2

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

nonn

Conjecture: For n >= 2, n! - a(n) is 1 or a prime, see A033933. - Amarnath Murthy, Mar 19 2002

a(n) is the largest prime divisor of (n!)! of the sequence A000197. - Stanislav Sykora, Jul 14 2014

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

Donovan Johnson, Table of n, a(n) for n = 2..400
R. G. Wilson v, Letter to N. J. A. Sloane, Oct. 1993

Cf. A000142, A007917.

PrevPrime[ n_Integer ] := Module[ {k = n - 1}, While[ ! PrimeQ[ k ], k-- ]; k ]; Table[ PrevPrime[ n! ], {n, 3, 25} ]
Join[{2},NextPrime[Range[3,30]!,-1]] (* Harvey P. Dale, Jan 24 2014 *)

More terms from Jud McCranie; also from Robert G. Wilson v, Jan 03 2001

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

A074967 a(n) = least k such that n^n - k is prime.

Original entry on oeis.org

1, 4, 5, 4, 7, 2, 3, 10, 33, 42, 19, 12, 17, 52, 59, 18, 65, 2, 51, 2, 23, 120, 35, 2, 63, 10, 39, 186, 7, 74, 47, 200, 53, 24, 19, 48, 333, 56, 57, 192, 127, 348, 63, 124, 213, 60, 359, 2, 213, 2, 387, 526, 269, 252, 863, 16, 131, 370, 503, 294, 83, 68, 317

Offset: 2

Zak Seidov, Oct 03 2002

nonn

Seiichi Manyama, Table of n, a(n) for n = 2..500

Cf. A033933, A074966.

PrimePrevDelta[n_]:=Module[{k},k=n-1;While[ !PrimeQ[k],k-- ];k=n-k]; lst={};Do[AppendTo[lst,PrimePrevDelta[n^n]],{n,2,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 11 2009 *)
lk[n_]:=Module[{nn=n^n},nn-NextPrime[nn,-1]]; Array[lk,70,2] (* Harvey P. Dale, Jan 20 2019 *)

a(n)=(x->x-precprime(x))(n^n) \\ Charles R Greathouse IV, Nov 25 2014

More terms from Robert G. Wilson v, Oct 04 2002

Offset corrected by R. J. Mathar, Jun 12 2009

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

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

Original entry on oeis.org

0, 1, 5, 7, 19, 19, 31, 17, 11, 17, 83, 67, 353, 227, 163, 59, 61, 113, 353, 31, 1447, 571, 389, 191, 337, 883, 101, 1823, 659, 709, 163, 1361, 439, 307, 1093, 1733, 2491, 1063, 1091, 1999, 1439, 109, 2753, 607, 2617, 269, 103, 2663, 337, 14447, 2221, 5471, 2887

Offset: 2

Zak Seidov, Sep 18 2002

nonn

For n=3,5,10,21,171,190,348, n! is an interprime, the average of two consecutive primes, see A053709. In general n! may be average of several pairs of primes, in which case the minimal distance is in the sequence. See also n^n and n!! as average of two primes in A075468 and A075410.

According to Goldbach's conjecture, a(n) always exists with a(n) = A047160(n!). - Jens Kruse Andersen, Jul 30 2014

			a(4)=5 because 4!=24 and 19 and 25 are primes with smallest distance 5 from 4!.

Vincenzo Librandi, Table of n, a(n) for n = 2..100

Cf. A033932, A033933, A047160, A053709, A075468, A075410.

smp[n_]:=Module[{m=1,nf=n!},While[!PrimeQ[nf+m]||!PrimeQ[nf-m],m=m+2];m]; Join[{0},Array[smp,60,3]] (* Harvey P. Dale, Apr 18 2014 *)

a(n) = {my (m=0); until (ok, ok = isprime(n!-m) && isprime(n!+m); if (!ok, m++);); return (m);} \\ Michel Marcus, Apr 19 2013

More terms from David Wasserman, Jan 17 2005

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

cp's OEIS Frontend

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

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

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A006990 Largest prime <= 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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A074967 a(n) = least k such that n^n - k is prime.

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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!.