%I #25 May 16 2013 18:50:00
%S 5,5,3,2,5,2,3,11,3,3,5,2,0,2,3,0,5,5,0,2,5,3,11,2,0,5,3,0,7,7,0,2,5,
%T 3,5,11,3,5,7,0,61,2,0,7,3,0,13,11,3,11,11,0,19,2,0,5,3,0,19,2,0,17,5,
%U 3,7,5,0,5,3,3,11,7,0,2,7,0,5,109
%N Smallest prime p such that n*(p-1)-1 and n*(p+1)+1 are both prime, or 0 if no such p exists.
%C Numbers n such that a(n) = 0: 13, 16, 19, 25, 28, 31, 40, 43, 46, 52, 55, 58, 61, 67, 73, 76, ...
%C a(n) = 0 if n = 1 mod 3 and none of the pairs {n-1, 3n+1}, {2n-1, 4n+1}, {n+1, n+2} have both members prime. On Dickson's conjecture "if" can be replaced with "if and only if". - _Charles R Greathouse IV_, May 07 2013
%C Smallest k > 1 such that n^k - n - 1 and n^k + n + 1 are both prime, or 0 if no such k exists: 0, 3, 2, 0, 2, 2, 0, 3, 3, 0, 4, 2, 0, 2, 3, 0, 2, 3, 0, 2, 2, 0,... - _Juri-Stepan Gerasimov_, May 09 2013
%H Charles R Greathouse IV, <a href="/A225302/b225302.txt">Table of n, a(n) for n = 1..10000</a>
%e a(1) = 5 because 1*5 - 1 - 1 = 3 and 1*5 + 1 + 1 = 7 are both prime,
%e a(2) = 5 because 2*5 - 2 - 1 = 7 and 2*5 + 2 + 1 = 13 are both prime,
%e a(3) = 3 because 3*3 - 3 - 1 = 5 and 3*3 + 3 + 1 = 13 are both prime.
%t a[n_] := Block[{p = 2}, If[n < 5, {5, 5, 3, 2}[[n]], If[Mod[n, 3] == 1, If[PrimeQ[2*n-1] && PrimeQ[4*n+1], 3, 0], While[! PrimeQ[n*(p - 1) -1] || ! PrimeQ[n*(p + 1) +1], p = NextPrime@p]; p]]]; Array[a, 80] (* _Giovanni Resta_, May 05 2013 *)
%o (PARI) a(n)=forprime(p=2,5,if(isprime(n*p-n-1) && isprime(n*p+n+1), return(p))); if(n%3==1,return(0)); forprime(p=7,,if(isprime(n*p-n-1) && isprime(n*p+n+1),return(p))) \\ _Charles R Greathouse IV_, May 07 2013
%Y Cf. A225063
%K nonn
%O 1,1
%A _Juri-Stepan Gerasimov_, May 05 2013
%E Corrected by _R. J. Mathar_, May 05 2013
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