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%I #17 Dec 05 2018 05:11:13
%S 0,0,1,2,3,2,2,2,2,3,4,4,3,2,3,3,5,6,3,3,4,4,5,7,4,3,4,2,5,7,4,4,5,4,
%T 5,7,4,5,8,2,5,7,5,5,6,6,4,7,4,5,9,3,5,9,4,6,6,5,5,7,3,3,7,3,6,8,5,4,
%U 8,4,5,8,4,4,5,3,5,8,6,3,6,4,5,12,5,5,5,3,6,8,5,4,8,4,4,8,4,6,9,5
%N Number of ways to write 2n-1 = p+q, where p is a prime, and both q and q+2 are practical numbers (A005153).
%C Conjecture: a(n)>0 for all n=3,4,...
%C The author has verified this for n up to 2*10^8. It is known that there are infinitely many practical numbers q with q+2 also practical.
%C Zhi-Wei Sun also made the following similar conjectures:
%C (1) Each odd number n>5 can be written as p+q with p and p+6 both prime and q practical. Also, any odd number n>3 not equal to 55 can be written as p+q with p and p+2 both prime and q practical.
%C (2) Each integer n>10 can be written as x+y (x,y>0) with 6x-1 and 6x+1 both prime, and y and y+6 both practical.
%C Also, any integer n>=6360 can be written as x+y (x,y>0) with 6x-1 and 6x+1 both prime, and y and y+2 both practical.
%H Zhi-Wei Sun, <a href="/A208243/b208243.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(14)=2 since 2*14-1=27=11+16=23+4, where 11 and 23 are primes, 16,16+2,4,4+2 are practical numbers.
%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[pr[2k]==True&&pr[2k+2]==True&&PrimeQ[2n-1-2k]==True,1,0],{k,1,n-1}]
%t Do[Print[n," ",a[n]],{n,1,100}]
%Y Cf. A000040, A005153, A199920.
%K nonn
%O 1,4
%A _Zhi-Wei Sun_, Jan 11 2013
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