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%I #14 Mar 20 2014 13:41:36
%S 0,0,0,0,1,1,2,2,4,3,3,3,2,4,2,5,3,4,5,1,5,3,6,7,5,9,3,7,5,4,7,5,9,5,
%T 5,4,2,4,2,5,4,6,7,5,9,6,9,8,7,10,8,10,6,7,6,6,7,6,5,6,7,5,5,6,7,8,7,
%U 10,11,12,11,7,6,9,10,8,7,6,7,5
%N Number of ways to write n = k + m with k > 0 and m > 0 such that pi(2*k) - pi(k) is prime and pi(2*m) - pi(m) is a square, where pi(x) denotes the number of primes not exceeding x.
%C Conjecture: a(n) > 0 for all n > 4, and a(n) = 1 only for n = 5, 6, 20.
%H Zhi-Wei Sun, <a href="/A239430/b239430.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>, preprint, arXiv:1402.6641, 2014.
%e a(5) = 1 since 5 = 4 + 1 with pi(2*4) - pi(4) = 4 - 2 = 2 prime and pi(2*1) - pi(1) = 1^2.
%e a(20) = 1 since 20 = 8 + 12 with pi(2*8) - pi(8) = 6 - 4 = 2 prime and pi(2*12) - pi(12) = 9 - 5 = 2^2.
%t SQ[n_]:=IntegerQ[Sqrt[n]]
%t s[n_]:=SQ[PrimePi[2n]-PrimePi[n]]
%t p[n_]:=PrimeQ[PrimePi[2n]-PrimePi[n]]
%t a[n_]:=Sum[If[p[k]&&s[n-k],1,0],{k,1,n-1}]
%t Table[a[n],{n,1,80}]
%Y Cf. A000040, A000290, A000720, A239428.
%K nonn
%O 1,7
%A _Zhi-Wei Sun_, Mar 20 2014
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