%I
%S 0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,3,3,2,3,4,4,3,4,5,4,3,5,6,4,3,
%T 5,7,5,4,6,8,4,3,5,8,4,2,4,8,5,3,4,7,4,3,5,7,3,2,4,6,5,4,4,7,5,4,5,7,
%U 4,2,4,7,5,3,4,6,4,4,6,6,3,2,5,6,4,4,5,7,5,5,7,8,2,2,6,8,5,3,4,7
%N Number of ways to write n = p+q with p prime or practical, and q4, q, q+4 all practical
%C Conjecture: a(n)>0 for all n>8.
%C ZhiWei Sun also made some similar conjectures, below are few examples.
%C (1) Each integer n>2 can be written as p+q with p prime or practical, and q and q+2 both practical.
%C (2) Any integer n>12 can be written as p+q with p prime or practical, and q8, q, q+8 all practical.
%C (3) The interval [n,2n) contains a practical number p with pn a triangular number.
%C (4) Any integer n>1 can be written as x^2+y (x,y>0) with 2x and 2xy both practical.
%C Note that if x>=y>0 with x practical then xy is also practical.
%H ZhiWei Sun, <a href="/A208246/b208246.txt">Table of n, a(n) for n = 1..20000</a>
%H ZhiWei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arxiv:1211.1588.
%e a(11)=1 since 11=3+8 with 3 prime, and 4, 8, 12 all practical.
%e a(12)=1 since 12=4+8 with 4, 8, 12 all practical.
%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<3Mod[n,2]+Con[n]==0)
%t a[n_]:=a[n]=Sum[If[pr[k]==True&&pr[k4]==True&&pr[k+4]==True&&(PrimeQ[nk]==Truepr[nk]==True),1,0],{k,1,n1}]
%t Do[Print[n," ",a[n]],{n,1,100}]
%Y Cf. A005153, A208243, A208244, A219842, 220272.
%K nonn
%O 1,13
%A _ZhiWei Sun_, Jan 11 2013
