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A209320 Number of ways to write 2n = p+q with p and q both prime, p+1 and q-1 both practical.

%I A209320 #15 Dec 05 2018 06:13:15
%S A209320 0,0,1,2,3,2,2,2,2,3,4,5,3,2,3,3,5,7,3,3,4,4,5,8,4,3,5,2,4,8,3,4,6,2,
%T A209320 4,7,3,4,7,2,4,9,4,4,9,5,3,9,3,5,8,3,4,10,4,6,8,5,4,14,2,4,8,2,6,10,4,
%U A209320 4,7,4,4,10,5,4,8,3,4,9,5,5,7,3,3,13,6,5,7,4,2,11,5,5,9,4,2,9,3,6,10,7
%N A209320 Number of ways to write 2n = p+q with p and q both prime, p+1 and q-1 both practical.
%C A209320 Conjecture: a(n)>0 for all n>2.
%C A209320 As p+q=(p+1)+(q-1), this unifies Goldbach's conjecture and its analog involving practical numbers.
%C A209320 The conjecture has been verified for n up to 10^7.
%H A209320 Zhi-Wei Sun, <a href="/A209320/b209320.txt">Table of n, a(n) for n = 1..10000</a>
%H A209320 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 A209320 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 A209320 a(8) = 2 since 2*8 = 3+13 = 11+5 with 3, 5, 11, 13 all prime and 3+1, 13-1, 11+1, 5-1 all practical.
%t A209320 f[n_]:=f[n]=FactorInteger[n]
%t A209320 Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
%t A209320 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 A209320 pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
%t A209320 a[n_]:=a[n]=Sum[If[PrimeQ[2n-Prime[k]]==True&&pr[Prime[k]+1]==True&&pr[2n-Prime[k]-1]==True,1,0],{k,1,PrimePi[2n-2]}]
%t A209320 Do[Print[n," ",a[n]],{n,1,100}]
%Y A209320 Cf. A005153, A002372, A045917, A208243, A208244, A208246, A208249, A209253, A209254, A209312, A219185, A219312, A219315.
%K A209320 nonn
%O A209320 1,4
%A A209320 _Zhi-Wei Sun_, Jan 19 2013