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%I #10 Apr 06 2014 22:15:45
%S 0,1,2,2,2,1,1,1,1,1,1,1,1,1,2,4,3,2,2,2,2,2,1,1,2,2,1,2,5,3,1,3,3,3,
%T 3,3,1,3,1,2,2,5,2,3,3,5,2,5,7,3,3,4,5,5,5,4,4,5,2,3,4,7,5,3,4,8,6,5,
%U 4,6,5,4,2,6,5,6,5,2,6,7
%N Number of ways to write n = k + m with k > 0 and m > 0 such that p(k)^2 - 2, p(m)^2 - 2 and p(p(m))^2 - 2 are all prime, where p(j) denotes the j-th prime.
%C Conjecture: a(n) > 0 for all n > 1.
%C This conjecture was motivated by the "Super Twin Prime Conjecture".
%C See A237414 for primes q with q^2 - 2 and p(q)^2 - 2 both prime.
%H Zhi-Wei Sun, <a href="/A237413/b237413.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;b81b9aa9.1402">Super Twin Prime Conjecture</a>, a message to Number Theory List, Feb. 6, 2014.
%H Z.-W. 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 7 = 6 + 1 with p(6)^2 - 2 = 13^2 - 2 = 167, p(1)^2 - 2 = 2^2 - 2 = 2 and p(p(1))^2 - 2 = p(2)^2 - 2 = 3^2 - 2 = 7 are all prime.
%e a(516) = 1 since 516 = 473 + 43 with p(473)^2 - 2 = 3359^2 - 2 = 11282879, p(43)^2 - 2 = 191^2 - 2 = 36479 and p(p(43))^2 - 2 = p(191)^2 - 2 = 1153^2 - 2 = 1329407 all prime.
%t pq[k_]:=PrimeQ[Prime[k]^2-2]
%t a[n_]:=Sum[If[pq[k]&&pq[n-k]&&pq[Prime[n-k]],1,0],{k,1,n-1}]
%t Table[a[n],{n,1,80}]
%Y Cf. A000040, A049002, A062326, A218829, A237348, A237367, A237414.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Feb 07 2014
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