A053710 -id:A053710 - cp's OEIS frontend

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

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

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

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

A350782 a(n) is the number of pairs of primes (p,q), p < q, such that (p + q)/2 = n!.

Original entry on oeis.org

1, 5, 18, 60, 315, 1615, 9928, 70437, 637504, 5829386, 64647125, 722750400

Offset: 3

Jon E. Schoenfield, Jan 15 2022

a(n) is the number of pairs of primes that are equidistant from n!.

Equivalently, a(n) is the number of positive integers d such that n! - d and n! + d are primes. For the smallest such d, iff there are no primes in the open interval (n! - d, n! + d), then n is a term in A053709, n! is a term in A053710, and d is a term in A053711.

			For n = 4, n! = 24, from which 5 pairs of primes are equidistant; in order of increasing distance, these are (19, 29), (17, 31), (11, 37), (7, 41), and (5, 43), so a(4) = 5. (A prime (23) lies between 19 and 29, so n=4 is not a term of A053709.)
For n = 5, n! = 120, from which 18 pairs of primes are equidistant: (113, 127), (109, 131), (103, 137), (101, 139), (89, 151), (83, 157), (73, 167), (67, 173), (61, 179), (59, 181), (47, 193), (43, 197), (41, 199), (29, 211), (17, 223), (13, 227), (11, 229), and (7, 233), so a(5) = 18. (The pair least distant from 120 is (113, 127), and there are no primes between 113 and 127, so n = 5 is a term of A053709, n! = 120 is a term of A053710, and d = 120 - 113 = 127 - 120 = 7 is a term of A053711.)

Cf. A002375, A053709, A053710, A053711.

Table[Length@IntegerPartitions[2n!,2,Prime@Range@PrimePi[2n!]],{n,3,9}] (* Giorgos Kalogeropoulos, Jan 16 2022 *)

from sympy import isprime, nextprime, factorial
def A350782(n):
    m, p, c = factorial(n), 3, 0
    while p <= m:
        if isprime(2*m-p):
            c += 1
        p = nextprime(p)
    return c # Chai Wah Wu, Jan 16 2022

a(n) = A002375(n!).

a(14) from Martin Ehrenstein, Jan 25 2022

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

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A053712 Lower balancing primes to prime-balanced factorials.

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A053713 Upper balancing primes to prime-balanced factorials.

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A350782 a(n) is the number of pairs of primes (p,q), p < q, such that (p + q)/2 = n!.

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