A238585 - OEIS

%I #7 Mar 01 2014 11:55:45

%S 0,0,0,1,1,0,1,2,2,1,1,1,3,2,3,2,2,3,1,5,1,1,3,2,4,5,2,4,3,4,1,4,5,3,

%T 4,6,3,2,2,2,2,1,8,1,3,4,7,2,5,3,2,2,4,7,4,3,2,3,5,7,5,3,6,6,5,3,4,5,

%U 2,2,2,3,7,2,3,7,3,4,10,3

%N Number of primes p < n with prime(p)^2 + (prime(n)-1)^2 prime.

%C Conjecture: (i) a(n) > 0 unless n divides 6, and a(n) = 1 only for n = 4, 5, 7, 10, 11, 12, 19, 21, 22, 31, 42, 44.

%C (ii) If n > 2 is not equal to 9, then prime(n)^2 + (prime(p) - 1)^2 is prime for some prime p < n.

%C (iii) For n > 3, there is a prime p < n with prime(p) + prime(n) + 1 prime. If n > 9 is not equal to 18, then prime(p)^2 + prime(n)^2 - 1 is prime for some prime p < n.

%H Zhi-Wei Sun, <a href="/A238585/b238585.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014.

%e a(7) = 1 since 3 and prime(3)^2 + (prime(7)-1)^2 = 5^2 + 16^2 = 281 are both prime.

%e a(44) = 1 since 23 and prime(23)^2 + (prime(44)-1)^2 = 83^2 + 192^2 = 43753 are both prime.

%t p[n_,k_]:=PrimeQ[k]&&PrimeQ[Prime[k]^2+(Prime[n]-1)^2]

%t a[n_]:=Sum[If[p[n,k],1,0],{k,1,n-1}]

%t Table[a[n],{n,1,80}]

%Y Cf. A000040, A232465, A238580.

%K nonn

%O 1,8

%A _Zhi-Wei Sun_, Mar 01 2014