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A208246 Number of ways to write n = p+q with p prime or practical, and q-4, q, q+4 all practical.

%I A208246 #30 Jul 22 2025 21:14:53
%S A208246 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 A208246 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 A208246 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 A208246 Number of ways to write n = p+q with p prime or practical, and q-4, q, q+4 all practical.
%C A208246 Conjecture: a(n)>0 for all n>8.
%C A208246 Zhi-Wei Sun also made some similar conjectures, below are few examples.
%C A208246 (1) Each integer n>3 can be written as p+q with p prime or practical, and q and q+2 both practical.
%C A208246 (2) Any integer n>12 can be written as p+q with p prime or practical, and q-8, q, q+8 all practical.
%C A208246 (3) The interval [n,2n) contains a practical number p with p-n a triangular number.
%C A208246 (4) Any integer n>1 can be written as x^2+y (x,y>0) with 2x and 2xy both practical.
%C A208246 Note that if x>=y>0 with x practical then xy is also practical.
%H A208246 Zhi-Wei Sun, <a href="/A208246/b208246.txt">Table of n, a(n) for n = 1..20000</a>
%H A208246 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures on representations involving primes</a>, arxiv:1211.1588 [math.NT], 2012-2017.
%e A208246 a(11)=1 since 11=3+8 with 3 prime, and 4, 8, 12 all practical.
%e A208246 a(12)=1 since 12=4+8 with 4, 8, 12 all practical.
%t A208246 f[n_]:=f[n]=FactorInteger[n]
%t A208246 Pow[n_,i_]:=Pow[n,i]=Part[Part[f[n],i],1]^(Part[Part[f[n],i],2])
%t A208246 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 A208246 pr[n_]:=pr[n]=n>0&&(n<3||Mod[n,2]+Con[n]==0)
%t A208246 a[n_]:=a[n]=Sum[If[pr[k]==True&&pr[k-4]==True&&pr[k+4]==True&&(PrimeQ[n-k]==True||pr[n-k]==True),1,0],{k,1,n-1}]
%t A208246 Do[Print[n," ",a[n]],{n,1,100}]
%Y A208246 Cf. A005153, A208243, A208244, A219842, A220272.
%K A208246 nonn
%O A208246 1,13
%A A208246 _Zhi-Wei Sun_, Jan 11 2013