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A222603 a(1)=1; for n>0, a(n+1) is the least practical number q>a(n) such that 2(a(n)+1)-q is practical.

%I A222603 #17 Dec 08 2018 11:22:04
%S A222603 1,2,4,6,8,12,18,20,24,30,32,36,42,54,56,60,66,78,80,84,90,104,120,
%T A222603 162,176,192,210,224,234,260,270,272,276,294,320,330,342,378,380,384,
%U A222603 390,392,396,414,416,420,450,462,464,468,476,486,510,512,522,546,594,620,630,702,704,714,726,728,744,750,798,800,810,812,816,920,924,930,966,968,972,980,990,992,1014,1040,1050,1088,1122,1232,1242,1254,1280,1290,1302,1316,1332,1350,1352,1380,1386,1458,1518,1520
%N A222603 a(1)=1; for n>0, a(n+1) is the least practical number q>a(n) such that 2(a(n)+1)-q is practical.
%C A222603 By a result of Melfi, each positive even number can be written as the sum of two practical numbers.
%C A222603 For a practical number p, define h(p) as the least practical number q>p such that 2(p+1)-q is practical. Construct a simple (undirected) graph H as follows: The vertex set of H is the set of all practical numbers, and for two vertices p and q>p there is an edge connecting p and q if and only if h(p)=q. Clearly H contains no cycle.
%C A222603 Conjecture: The graph H constructed above is connected and hence it is a tree.
%H A222603 Zhi-Wei Sun, <a href="/A222603/b222603.txt">Table of n, a(n) for n = 1..10000</a>
%H A222603 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>].
%e A222603 a(4)=6 since 2(a(3)+1)=10=6+4 with 4 and 6 both practical, and 6>a(3)=4.
%t A222603 f[n_]:=f[n]=FactorInteger[n]
%t A222603 Pow[n_,i_]:=Pow[n,i]=Part[Part[f[n],i],1]^(Part[Part[f[n],i],2])
%t A222603 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 A222603 pr[n_]:=pr[n]=n>0&&(n<3||Mod[n,2]+Con[n]==0)
%t A222603 k=1
%t A222603 n=1
%t A222603 Do[If[m==1,Print[n," ",1]];If[m==k,n=n+1;Do[If[pr[2j]==True&&pr[2m+2-2j]==True,k=2j;Print[n," ",2j];Goto[aa]],{j,Ceiling[(m+1)/2],m}]];
%t A222603 Label[aa];Continue,{m,1,1000}]
%Y A222603 Cf. A005153, A222532, A163846, A163847, A222566.
%K A222603 nonn
%O A222603 1,2
%A A222603 _Zhi-Wei Sun_, Feb 26 2013