|
%I #11 Feb 08 2014 10:06:27
%S 0,0,0,0,0,1,2,4,4,3,2,3,3,3,5,3,1,4,5,5,7,4,1,2,1,1,1,1,1,3,6,7,8,8,
%T 8,8,8,9,11,11,11,11,9,7,7,4,1,2,1,2,3,5,7,10,14,14,14,10,6,10,14,16,
%U 19,16,13,12,11,10,7,6,5,3,3,4,3,6,9,13,17,18
%N Number of ordered ways to write n = k + m (0 < k <= m) with pi(k) + pi(m) - 2 prime, where pi(.) is given by A000720.
%C Conjecture: (i) a(n) > 0 for all n > 5.
%C (ii) Any integer n > 23 can be written as k + m (k > 0 and m > 0) with pi(k) + pi(m) prime. Also, each integer n > 25 can be written as k + m (k > 0 and m > 0) with pi(k) + pi(m) - 1 prime.
%H Zhi-Wei Sun, <a href="/A237496/b237496.txt">Table of n, a(n) for n = 1..3000</a>
%e a(6) = 1 since 6 = 3 + 3 with pi(3) + pi(3) - 2 = 2 + 2 - 2 = 2 prime.
%e a(17) = 1 since 17 = 2 + 15 with pi(2) + pi(15) - 2 = 1 + 6 - 2 = 5 prime.
%e a(99) = 1 since 99 = 1 + 98 with pi(1) + pi(98) - 2 = 0 + 25 - 2 = 23 prime.
%t PQ[n_]:=n>0&&PrimeQ[n]
%t p[k_,m_]:=PQ[PrimePi[k]+PrimePi[m]-2]
%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, A232465, A237284, A237291, A237453, A237497.
%K nonn
%O 1,7
%A _Zhi-Wei Sun_, Feb 08 2014
|
