A136019 - OEIS

%I #14 May 20 2021 16:33:31

%S 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,

%T 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,

%U 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

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

%C The associated prime(k) are in A136020.

%H Robert Israel, <a href="/A136019/b136019.txt">Table of n, a(n) for n = 1..10000</a>

%e a(1)=3 because 3 is smallest prime of the form (p+2)/3; in this case prime(k)=7.

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

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

%p N:= 10^5: # to allow prime(k) <= N

%p Primes:= select(isprime,[2,seq(2*i+1,i=1..floor((N-1)/2))]):

%p f:= proc(t,n)

%p   local s;

%p   s:= (t+2*n)/(1+2*n);

%p   type(s,integer) and isprime(s)

%p end proc:

%p for n from 1 do

%p   p:= ListTools:-SelectFirst(f, Primes,n);

%p   if p = NULL then break fi;

%p   A[n]:= (p+2*n)/(1+2*n);

%p od:

%p seq(A[i],i=1..n-1); # _Robert Israel_, Sep 08 2014

%t a = {}; Do[k = 1; While[ !PrimeQ[(Prime[k] + 2n)/(2n + 1)], k++ ]; AppendTo[a, (Prime[k] + 2n)/(2n + 1)], {n, 1, 200}]; a

%t 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 *)

%o (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

%Y Cf. A136020, A136026, A136027.

%K nonn,easy

%O 1,1

%A _Artur Jasinski_, Dec 10 2007

%E Edited by _R. J. Mathar_, May 17 2009