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A209254 Number of ways to write 2n-1 = p+q with q practical, p and p^4+q^4 both prime.

%I A209254 #14 Dec 05 2018 05:11:00
%S A209254 0,1,1,2,2,2,3,1,2,2,3,2,4,3,1,3,1,1,4,2,5,5,1,4,1,2,4,3,1,6,3,4,4,5,
%T A209254 1,6,7,2,4,3,4,2,4,5,1,2,3,7,5,2,4,8,4,6,5,1,2,2,3,8,3,1,5,6,2,4,7,4,
%U A209254 8,4,2,7,6,3,4,3,1,6,6,1,7,6,2,8,9,5,7,3,3,10,7,3,9,14,1,9,4,3,4,6
%N A209254 Number of ways to write 2n-1 = p+q with q practical, p and p^4+q^4 both prime.
%C A209254 Conjecture: a(n)>0 for all n>1.
%C A209254 Zhi-Wei Sun also conjectured that any odd integer greater than one can be written as p+q with q practical, and p and p^2+q^2 both prime. This is a refinement of Ming-Zhi Zhang's problem related to A036468.
%H A209254 Zhi-Wei Sun, <a href="/A209254/b209254.txt">Table of n, a(n) for n = 1..10000</a>
%H A209254 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 A209254 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 A209254 a(8)=1 since 2*8-1=11+4 with 4 practical, 11 and 11^4+4^4=14897 both prime.
%t A209254 f[n_]:=f[n]=FactorInteger[n]
%t A209254 Pow[n_,i_]:=Pow[n,i]=Part[Part[f[n],i],1]^(Part[Part[f[n],i],2])
%t A209254 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 A209254 pr[n_]:=pr[n]=n>0&&(n<3||Mod[n,2]+Con[n]==0)
%t A209254 a[n_]:=a[n]=Sum[If[pr[2n-1-Prime[k]]==True&&PrimeQ[Prime[k]^4+(2n-1-Prime[k])^4]==True,1,0],{k,1,PrimePi[2n-1]}]
%t A209254 Do[Print[n," ",a[n]],{n,1,100}]
%Y A209254 Cf. A000040, A005153, A208243, A208244, A208246, A209253.
%K A209254 nonn
%O A209254 1,4
%A A209254 _Zhi-Wei Sun_, Jan 14 2013