A136019 Smallest prime of the form (prime(k)+2*n)/(2*n+1), any k.

3, 3, 5, 3, 3, 5, 3, 7, 11, 3, 3, 5, 5, 3, 11, 3, 3, 5, 3, 3, 5, 5, 7, 5, 3, 3, 7, 5, 13, 7, 3, 3, 5, 3, 13, 5, 3, 7, 5, 3, 3, 13, 5, 3, 7, 5, 3, 5, 3, 7, 7, 3, 7, 11, 3, 3, 5, 11, 3, 7, 7, 3, 5, 11, 3, 13, 3, 7, 5, 3, 7, 11, 7, 13, 7, 3, 3, 11, 23, 7, 5, 3, 31, 5, 13, 3, 5, 5, 3, 7, 3, 13, 7, 3, 3, 5, 7

Offset: 1

Author

Artur Jasinski, Dec 10 2007

Keywords

• nonn
• easy

Comments

The associated prime(k) are in A136020.

Examples

			a(1)=3 because 3 is smallest prime of the form (p+2)/3; in this case prime(k)=7.
a(2)=3 because 3 is smallest prime of the form (p+4)/5; in this case prime(k)=11.
a(3)=5 because 5 is smallest prime of the form (p+6)/7; in this case prime(k)=29.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A136020, A136026, A136027.

Programs

Maple

N:= 10^5: # to allow prime(k) <= N
Primes:= select(isprime,[2,seq(2*i+1,i=1..floor((N-1)/2))]):
f:= proc(t,n)
  local s;
  s:= (t+2*n)/(1+2*n);
  type(s,integer) and isprime(s)
end proc:
for n from 1 do
  p:= ListTools:-SelectFirst(f, Primes,n);
  if p = NULL then break fi;
  A[n]:= (p+2*n)/(1+2*n);
od:
seq(A[i],i=1..n-1); # Robert Israel, Sep 08 2014

Mathematica

a = {}; Do[k = 1; While[ !PrimeQ[(Prime[k] + 2n)/(2n + 1)], k++ ]; AppendTo[a, (Prime[k] + 2n)/(2n + 1)], {n, 1, 200}]; a
sp[n_]:=Module[{k=1},While[!PrimeQ[(Prime[k]+2n)/(2n+1)],k++];(Prime[ k]+2n)/(2n+1)]; Array[sp,100] (* Harvey P. Dale, May 20 2021 *)

PARI

a(n)=my(N=2*n,k=0,t);forprime(p=2,default(primelimit),k++;t=(p+N)/(N+1);if(denominator(t)==1&isprime(t),return(t))) \\ Charles R Greathouse IV, Jun 16 2011

Extensions

Edited by R. J. Mathar, May 17 2009