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

1,1

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

Links

Table of n, a(n) for n=1..9.

Example

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)

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}]

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).

Sequence in context: A241436 A018107 A360882 * A102772 A018005 A080522

Adjacent sequences: A053706 A053707 A053708 * A053710 A053711 A053712

Keyword

nonn,more

Author

Labos Elemer, Feb 10 2000

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

Status

approved