A222580
Number of ways to write n=p_m-p_{m-1}+...+(-1)^{m-k}p_k with k
1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 3, 2, 3, 2, 2, 2, 3, 1, 2, 2, 3, 2, 2, 1, 2, 3, 3, 1, 1, 2, 4, 2, 1, 3, 2, 2, 2, 3, 3, 2, 2, 2, 2, 6, 1, 2, 3, 4, 2, 3, 3, 2, 3, 4, 2, 4, 2, 1, 1, 4, 3, 4, 2, 4, 1, 3, 3, 2, 4, 4, 2, 3, 2, 3, 3, 3, 3, 2, 5, 1, 3, 4, 7, 4, 2, 3, 2, 1, 5, 2, 4, 2, 7, 3, 3, 3, 4, 5, 6
Offset: 1
Keywords
Examples
a(9)=2 since 9=11-7+5=19-17+13-11+7-5+3 with 12, 4, 20, 2 all practical.
a(806)=1 since 806=p_{358}-p_{357}+...+p_{150}-p_{149} with p_{358}=2411<=3*806=2418, and 2412 and p_{149}-1=858 are both practical.
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.
- Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.
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] a[n_]:=a[n]=Sum[If[pp[j]==True&&pq[i+1]==True&&s[j]-(-1)^(j-i)*s[i]==n,1,0],{j,PrimePi[n]+1,PrimePi[3n]},{i,0,j-2}] Table[a[n],{n,1,100}]
Comments