A102723 Smallest prime a(n) such that a(n)-x and a(n)+x, for x=1 to n, are all composite.

5, 23, 23, 53, 53, 211, 211, 211, 211, 211, 211, 1847, 1847, 2179, 2179, 2179, 2179, 3967, 3967, 16033, 16033, 16033, 16033, 24281, 24281, 24281, 24281, 24281, 24281, 38501, 38501, 38501, 38501, 38501, 38501, 38501, 38501, 38501, 38501, 58831

Offset: 1

Author

Ray G. Opao, Feb 06 2005

Keywords

• nonn

Comments

a(2n+1)=a(2n). - Robert G. Wilson v, Feb 22 2005

Using Dirichlet's theorem, Sierpiński (1948) proved that a(n) exists for all n > 0. He noted that a(n) is a non-twin prime (A007510), except for a(1) = 5. - Jonathan Sondow, Oct 27 2017

Links

• David A. Corneth, Table of n, a(n) for n = 1..479 (first 97 terms from Harvey P. Dale)
• W. Sierpiński, Remarque sur la répartition des nombres premiers, Colloq. Math., 1 (1948), 193-194.

Crossrefs

Cf. A007510, A023186.

Programs

Mathematica

f[n_] := Block[{k = 1}, While[ Union[ PrimeQ /@ Sort[ Flatten[ Table[{Prime[k] - i, Prime[k] + i}, {i, n}]]]] != {False}, k++ ]; Prime[k]]; Table[ f[n], {n, 40}] (* Robert G. Wilson v, Feb 22 2005 *)
cmpgap[n_]:=Module[{p=Prime[n]},Min[p-NextPrime[p,-1],NextPrime[p]-p]]; Module[{nn=10000,prs},prs=Table[{Prime[n],cmpgap[n]},{n,nn}];Table[ SelectFirst[ prs,#[[2]]>=k&],{k,2,50}]][[All,1]] (* Harvey P. Dale, Oct 15 2021 *)

Extensions

a(12)-a(40) from Robert G. Wilson v, Feb 22 2005