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

3, 5, 7, 5, 7, 11, 19, 11, 11, 17, 19, 17, 17, 23, 19, 23, 23, 31, 31, 41, 23, 41, 31, 47, 29, 47, 41, 59, 53, 59, 47, 59, 59, 79, 41, 83, 59, 79, 47, 83, 71, 83, 53, 83, 47, 103, 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

Offset: 1

Author

Zhi-Wei Sun, Feb 25 2013

Keywords

• nonn

Comments

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.

Examples

			a(6)=11 since 6=11-7+5-3 with 12 and 2 both practical;
a(7)=19 since 7=19-17+13-11+7-5+3-2 with 20 and 1 both practical;
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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, On functions taking only prime values, arXiv:1202.6589.

Crossrefs

Cf. A000040, A005153, A210479.

Programs

Mathematica

f[n_]:=f[n]=FactorInteger[n]
Pow[n_,i_]:=Pow[n,i]=Part[Part[f[n],i],1]^(Part[Part[f[n],i],2])
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}]
pr[n_]:=pr[n]=n>0&&(n<3||Mod[n,2]+Con[n]==0)
pp[k_]:=pp[k]=pr[Prime[k]+1]==True
pq[k_]:=pq[k]=pr[Prime[k]-1]==True
s[0_]:=0
s[n_]:=s[n]=Prime[n]-s[n-1]
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}];
Print[m," ",counterexample];Label[aa];Continue,{m,1,100}]