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%I #8 Apr 06 2014 22:16:57
%S 0,0,0,2,2,1,2,1,1,0,1,4,3,1,1,1,3,2,6,2,2,2,4,1,1,3,3,3,1,3,3,7,4,5,
%T 4,6,5,5,3,3,3,5,7,4,1,6,7,7,5,4,1,2,3,5,5,6,8,8,6,4,9,8,6,3,7,9,6,5,
%U 4,10,5,4,6,6,4,9,10,6,8,7
%N a(n) = |{0 < k <= n/2: pi(k*(n-k)) is prime}|, where pi(.) is given by A000720.
%C Conjecture: a(n) > 0 for all n > 10, and a(n) = 1 for no n > 51. Moreover, for any integer n > 10, there is a positive integer k < n with 2*k + 1 and pi(k*(n-k)) both prime.
%H Zhi-Wei Sun, <a href="/A237497/b237497.txt">Table of n, a(n) for n = 1..3000</a>
%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(6) = 1 since 6 = 1 + 5 with pi(1*5) = 3 prime.
%e a(8) = 1 since 8 = 2 + 6 with pi(2*6) = pi(12) = 5 prime.
%e a(25) = 1 since 25 = 4 + 21 with pi(4*21) = pi(84) = 23 prime.
%e a(51) = 1 since 51 = 14 + 37 with pi(14*37) = pi(518) = 97 prime.
%t p[k_,m_]:=PrimeQ[PrimePi[k*m]]
%t a[n_]:=Sum[If[p[k,n-k],1,0],{k,1,n/2}]
%t Table[a[n],{n,1,80}]
%Y Cf. A000040, A000720, A237284, A237291, A237453, A237496.
%K nonn
%O 1,4
%A _Zhi-Wei Sun_, Feb 08 2014
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