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A222579 Least prime p_m with p_m+1 practical such that n=p_m -p_{m-1}+...+(-1)^{m-k}p_k for some 0 with p_k-1 practical.

%I A222579 #13 Feb 25 2013 09:50:46
%S A222579 3,5,7,5,7,11,19,11,11,17,19,17,17,23,19,23,23,31,31,41,23,41,31,47,
%T A222579 29,47,41,59,53,59,47,59,59,79,41,83,59,79,47,83,71,83,53,83,47,103,
%U A222579 79,107,53,103,59,103,89,103,71,131,79,127,103,131,79,127,83,149,71,127,89,127,107,127,79,191,83,149,107,197,83,149,131,167,139,149,103,149,89,149,103,167,127,179,149,167,107,167,139,167,107,179,103,179
%N A222579 Least prime p_m with p_m+1 practical such that n=p_m -p_{m-1}+...+(-1)^{m-k}p_k for some 0<k<m with p_k-1 practical.
%C A222579 Conjecture: a(n)<=3n for all n>0. Moreover, a(2n-1)/(2n-1) and a(2n)/(2n) have limits 1 and 2 respectively, as n tends to the infinity.
%H A222579 Zhi-Wei Sun, <a href="/A222579/b222579.txt">Table of n, a(n) for n = 1..10000</a>
%H A222579 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1202.6589">On functions taking only prime values</a>, arXiv:1202.6589.
%e A222579 a(6)=11 since 6=11-7+5-3 with 12 and 2 both practical;
%e A222579 a(7)=19 since 7=19-17+13-11+7-5+3-2 with 20 and 1 both practical;
%e A222579 a(806)=p_{358}=2411 since 806=p_{358}-p_{357}+...+p_{150}-p_{149} with p_{358}+1=2412 and p_{149}-1=858 both practical. Note that a(806)/806 is about 2.9913.
%t A222579 f[n_]:=f[n]=FactorInteger[n]
%t A222579 Pow[n_,i_]:=Pow[n,i]=Part[Part[f[n],i],1]^(Part[Part[f[n],i],2])
%t A222579 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 A222579 pr[n_]:=pr[n]=n>0&&(n<3||Mod[n,2]+Con[n]==0)
%t A222579 pp[k_]:=pp[k]=pr[Prime[k]+1]==True
%t A222579 pq[k_]:=pq[k]=pr[Prime[k]-1]==True
%t A222579 s[0_]:=0
%t A222579 s[n_]:=s[n]=Prime[n]-s[n-1]
%t A222579 Do[Do[If[pp[j]==True&&pq[i+1]==True&&s[j]-(-1)^(j-i)*s[i]==m,Print[m," ",Prime[j]];Goto[aa]],{j,PrimePi[m]+1,PrimePi[3m]},{i,0,j-2}];
%t A222579 Print[m," ",counterexample];Label[aa];Continue,{m,1,100}]
%Y A222579 Cf. A000040, A005153, A210479.
%K A222579 nonn
%O A222579 1,1
%A A222579 _Zhi-Wei Sun_, Feb 25 2013