cp's OEIS Frontend

A198472 a(n)=q(n) if 4 | q(n)-2, and a(n)=q(n)/2 if 4 | q(n), where q(n) is the least practical number q>n with 2(n+1)-q practical.

%I A198472 #21 Dec 08 2018 11:21:07
%S A198472 2,2,2,6,6,4,4,6,6,8,6,18,8,18,8,18,18,10,10,12,12,14,12,30,14,30,14,
%T A198472 30,30,16,16,18,18,20,18,42,20,42,20,42,42,54,24,24,28,54,24,28,30,54,
%U A198472 28,32,54,28,28,30,30,32,30,66,32,66,32,66,66,78,36,36,40,78,36,40,42,78,40,44,78,40,40,42,42,44,42,90,44,90,44,90,90,52,48,48,50,50,48,52,50,54,50,56
%N A198472 a(n)=q(n) if 4 | q(n)-2, and a(n)=q(n)/2 if 4 | q(n), where q(n) is the least practical number q>n with 2(n+1)-q practical.
%C A198472 Conjecture: If b(1)>=4 is an integer and b(k+1)=a(b(k)) for k=1,2,3,..., then b(n)=4 for some n>0.
%C A198472 This conjecture has the same flavor as the Collatz conjecture.
%H A198472 Zhi-Wei Sun, <a href="/A198472/b198472.txt">Table of n, a(n) for n = 1..10000</a>
%H A198472 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 A198472 a(20)=12 since 2(20+1)=24+18 with 24 and 18 both practical.
%t A198472 f[n_]:=f[n]=FactorInteger[n]
%t A198472 Pow[n_,i_]:=Pow[n,i]=Part[Part[f[n],i],1]^(Part[Part[f[n],i],2])
%t A198472 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 A198472 pr[n_]:=pr[n]=n>0&&(n<3||Mod[n,2]+Con[n]==0)
%t A198472 Do[Do[If[pr[2k]==True&&pr[2n+2-2k]==True,Print[n," ",2k/(1+Mod[k-1,2])];Goto[aa]],{k,Ceiling[(n+1)/2],n}];
%t A198472 Label[aa];Continue,{n,1,100}]
%o A198472 (PARI) A198472(n) = forstep(q=n+++bittest(n,0),9e9,2, is_A005153(q) && is_A005153(2*n-q) && return(if(q%4,q,q\2))) \\ _M. F. Hasler_, Feb 27 2013
%Y A198472 Cf. A005153, A163846, A163847, A222603, A222532, A222566.
%K A198472 nonn
%O A198472 1,1
%A A198472 _Zhi-Wei Sun_, Feb 27 2013