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A210533 Number of ways to write 2n = x+y (x,y>0) with x-1 and x+1 both prime, and x and x^3+y^3 both practical.

%I A210533 #15 Dec 08 2018 11:21:30
%S A210533 0,0,1,2,2,2,3,3,3,4,4,4,4,4,4,5,1,5,3,5,5,5,3,6,5,6,6,6,3,6,2,7,7,6,
%T A210533 6,7,1,6,8,8,3,8,4,6,8,7,4,8,4,8,8,6,5,8,5,6,9,7,3,9,6,8,9,8,5,9,3,7,
%U A210533 9,7,5,9,2,7,9,7,4,10,4,8,10,8,5,10,8,7,10,10,6,10,4,9,11,8,7,11,6,11,12,11
%N A210533 Number of ways to write 2n = x+y (x,y>0) with x-1 and x+1 both prime, and x and x^3+y^3 both practical.
%C A210533 Conjecture: a(n)>0 for all n>2. Moreover, for each m=2,3,4,... any sufficiently large even integer can be written as x+y (x,y>0) with x-1 and x+1 both prime, and x and x^m+y^m both practical.
%H A210533 Zhi-Wei Sun, <a href="/A210533/b210533.txt">Table of n, a(n) for n = 1..10000</a>
%H A210533 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 A210533 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 A210533 a(17)=1 since 2*17=12+22 with 11 and 13 both prime, and 12 and 12^3+22^3=12376 both practical.
%t A210533 f[n_]:=f[n]=FactorInteger[n]
%t A210533 Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
%t A210533 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 A210533 pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
%t A210533 a[n_]:=a[n]=Sum[If[PrimeQ[2k-1]==True&&PrimeQ[2k+1]==True&&pr[2k]==True&&pr[(2k)^3+(2n-2k)^3]==True,1,0],{k,1,n-1}]
%t A210533 Do[Print[n," ",a[n]],{n,1,100}]
%Y A210533 Cf. A005153, A210528, A210531, A208243, A208244, A208246, A208249, A209236, A209253, A209254, A209312, A209315, A209320.
%K A210533 nonn
%O A210533 1,4
%A A210533 _Zhi-Wei Sun_, Jan 28 2013