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%I #21 Dec 05 2018 06:13:07
%S 0,0,0,0,1,1,1,0,1,2,1,2,1,2,3,2,2,2,3,1,3,4,2,2,2,3,4,3,1,3,3,1,4,5,
%T 3,3,3,2,5,4,1,3,5,2,5,4,3,4,5,2,5,5,2,4,5,3,6,5,5,5,2,3,6,5,2,3,4,3,
%U 6,5,4,4,4,5,6,6,4,5,4,3,6,8,2,2,5,6,7,6,2,6,2,4,7,6,4,3,6,3,5,5
%N Number of ways to write 2n-1 = p+q with q practical, p and q-p both prime.
%C Conjecture: a(n)>0 for all n>8.
%C This has been verified for n up to 10^7.
%C As p+q=2p+(q-p), the conjecture implies Lemoine's conjecture related to A046927.
%C Zhi-Wei Sun also conjectured that any integer n>2 can be written as p+q, where p is a prime, one of q and q+1 is prime and another of q and q+1 is practical.
%H Zhi-Wei Sun, <a href="/A209315/b209315.txt">Table of n, a(n) for n = 1..10000</a>
%H G. Melfi, <a href="http://dx.doi.org/10.1006/jnth.1996.0012">On two conjectures about practical numbers</a>, J. Number Theory 56 (1996) 205-210 [<a href="http://www.ams.org/mathscinet-getitem?mr=1370203">MR96i:11106</a>].
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588 [math.NT], 2012-2017.
%e a(9)=1 since 2*9-1=5+12 with 12 practical, 5 and 12-5 both prime.
%t f[n_]:=f[n]=FactorInteger[n]
%t Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
%t Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}]
%t pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
%t a[n_]:=a[n]=Sum[If[PrimeQ[p]==True&&pr[2n-1-p]==True&&PrimeQ[2n-1-2p]==True,1,0],{p,1,n-1}]
%t Do[Print[n," ",a[n]],{n,1,100}]
%Y Cf. A005153, A046927, A208243, A208244, A208246, A208249, A209253, A209254, A209312, A219185.
%K nonn
%O 1,10
%A _Zhi-Wei Sun_, Jan 19 2013
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