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%I #47 Jan 18 2022 10:14:56
%S 3,5,10,21,171,190,348,1638,3329
%N Prime balanced factorials: numbers k such that k! is the mean of its 2 closest primes.
%C Also, the integers k such that A033932(k) = A033933(k).
%C k! is an interprime, i.e., the average of two successive primes.
%C The difference between k! and any of its two closest primes must be 1 or exceed k. - _Franklin T. Adams-Watters_
%C Larger terms may involve probable primes. - _Hans Havermann_, Aug 14 2014
%e For the 1st term, 3! is in the middle between its closest prime neighbors 5 and 7.
%e For the 2nd term, 5! is in the middle between its closest prime neighbors 113 and 127.
%e From _Jon E. Schoenfield_, Jan 14 2022: (Start)
%e 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):
%e .
%e n k d
%e - ---- ----
%e 1 3 1
%e 2 5 7
%e 3 10 11
%e 4 21 31
%e 5 171 397
%e 6 190 409
%e 7 348 1657
%e 8 1638 2131
%e 9 3329 7607
%e (End)
%p 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:
%t 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}]
%Y Cf. A053711 (distances), A033932, A033933, A006990, A037151, A006562, A053710, A075275.
%Y Cf. A075409 (smallest m such that n!-m and n!+m are both primes).
%K nonn,more
%O 1,1
%A _Labos Elemer_, Feb 10 2000
%E a(5)-a(6) from _Jud McCranie_, Jul 04 2000
%E a(7) from _Robert G. Wilson v_, Sep 17 2002
%E a(8) from _Donovan Johnson_, Mar 23 2008
%E a(9) from _Hans Havermann_, Aug 14 2014