A089487 - OEIS

%I #19 Jun 29 2024 07:16:31

%S 11,7,17,11,5,7,41,23,17,23,13,31,53,17,17,17,29,19,19,5,13,13,149,41,

%T 11,11,5,137,19,5,7,23,59,13,29,11,11,13,11,59,23,13,11,5,41,41,19,19,

%U 71,31,23,11,31,41,41,47,41,7,11,53,17,29,19,53,5,101,13

%N The third-smallest prime of the form (p-prime(n))/(prime(n)-1), where p is also prime.

%H Amiram Eldar, <a href="/A089487/b089487.txt">Table of n, a(n) for n = 1..1000</a>

%e For n = 1, prime(n) = 2, and the ratios generated are (3-2)/1 = 1 (not prime), (5-2)/1 = 3 (prime, first), (7-2)/1 = 5 (prime, second), (11-2)/1 = 9 (not prime) and (13-2)/1 = 11 (prime, third and selected a(1)).

%p A089487 := proc(n) local ct,q,p ;

%p         ct := 0 ; q := ithprime(n) ; p := nextprime(q) ;

%p         while true do

%p                 while true do

%p                         if type( (p-q)/(q-1),'integer') then if isprime( (p-q)/(q-1)) then break;  end if;

%p                         end if;

%p                         p := nextprime(p) ;

%p                 end do:

%p                 ct := ct+1 ;

%p                 if ct = 3 then return (p-q)/(q-1); end if;

%p                 p := nextprime(p) ;

%p         end do:

%p end proc:

%p seq(A089487(n),n=1..44) ; # _R. J. Mathar_, Dec 06 2010

%t a[n_, r_] := Module[{p = Prime[n], q, rat, c = 0}, q = p; While[c < r, q = NextPrime[q]; If[PrimeQ[rat = (q - p)/(p - 1)], c++]]; rat]; Table[a[n, 3], {n, 1, 100}] (* _Amiram Eldar_, Jun 29 2024 *)

%o (PARI) /* r is the occurrence desired 1=first,2=second etc. */ diff2sqp2(n,r) = { forprime(q=2,n, c=0; forprime(p=q+1,n, y=(p-q)/(q-1); if(y==floor(y), if(isprime(y),c++; if(c==r, print1(y",");break)) ) ) ) }

%Y Cf. A089452, A089486.

%K easy,nonn

%O 1,1

%A _Cino Hilliard_, Dec 28 2003

%E Edited and corrected by _D. S. McNeil_, Dec 06 2010

%E More terms from _Amiram Eldar_, Jun 29 2024