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

Author

Labos Elemer, Feb 10 2000

Keywords

• nonn
• more

Comments

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

Examples

			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)
		

Crossrefs

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

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

Programs

Maple

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:

Mathematica

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}]

Extensions

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