cp's OEIS Frontend

A237367 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that 2*k - 1, prime(k)^2 - 2 and prime(m)^2 - 2 are all prime.

%I A237367 #12 Feb 07 2014 03:58:20
%S A237367 0,0,1,2,3,3,3,3,2,3,2,4,3,5,2,6,3,6,2,4,3,4,2,4,3,4,4,4,3,8,3,4,5,6,
%T A237367 6,5,6,5,5,3,4,7,5,6,3,7,3,3,5,4,5,6,5,8,10,4,5,11,6,3,6,5,5,5,6,5,8,
%U A237367 4,3,5,6,5,1,7,6,3,3,5,6,4
%N A237367 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that 2*k - 1, prime(k)^2 - 2 and prime(m)^2 - 2 are all prime.
%C A237367 Conjecture: (i) a(n) > 0 for all n > 2, and a(n) = 1 only for n = 3, 73, 81, 534.
%C A237367 (ii) Any integer n > 2 can be written as k + m with k > 0 and m > 0 such that 2*k - 1, prime(k) + k*(k-1) and prime(m) + m*(m-1) are all prime.
%C A237367 (iii) Every n = 9, 10, ... can be written as k + m with k > 0 and m > 0 such that 6*k - 1, prime(k) + 2*k and prime(m) + 2*m are all prime.
%C A237367 Clearly, part (i) of this conjecture implies that there are infinitely many primes p with p^2 - 2 also prime. Similar comments apply to parts (ii) and (iii).
%H A237367 Zhi-Wei Sun, <a href="/A237367/b237367.txt">Table of n, a(n) for n = 1..10000</a>
%e A237367 a(3) = 1 since 3 = 2 + 1 with 2*2 - 1 = 3, prime(2)^2 - 2 = 3^2 - 2 = 7 and prime(1)^2 - 2 = 2^2 - 2 = 2 all prime.
%e A237367 a(73) = 1 since 73 = 55 + 18 with 2*55 - 1 = 109, prime(55)^2 - 2 = 257^2 - 2 = 66047 and prime(18)^2 - 2 = 61^2 - 2 = 3719 all prime.
%e A237367 a(81) = 1 since 81 = 34 + 47 with 2*34 - 1 = 67, prime(34)^2 - 2 = 139^2 - 2 = 19319 and prime(47)^2 - 2 = 211^2 - 2 = 44519 all prime.
%e A237367 a(534) = 1 since 534 = 100 + 434 with 2*100 - 1 = 199, prime(100)^2 - 2 = 541^2 - 2 = 292679 and prime(434)^2 - 2 = 3023^2 - 2 = 9138527 all prime.
%t A237367 pq[k_]:=PrimeQ[Prime[k]^2-2]
%t A237367 a[n_]:=Sum[If[PrimeQ[2k-1]&&pq[k]&&pq[n-k],1,0],{k,1,n-1}]
%t A237367 Table[a[n],{n,1,80}]
%Y A237367 Cf. A000040, A049002, A062326, A076297, A218829, A230502, A233204, A237348.
%K A237367 nonn
%O A237367 1,4
%A A237367 _Zhi-Wei Sun_, Feb 07 2014