A257924 Primes p with p-1, p+1, prime(p)-1 and prime(p)+1 all practical.
3, 7, 31, 89, 199, 8009, 11551, 20129, 23549, 38609, 47501, 67231, 96221, 97001, 103409, 111871, 120473, 131071, 143261, 146681, 168869, 174761, 183091, 193951, 196181, 208279, 208961, 219727, 229769, 237691, 238519, 240641, 247759, 270271, 290249, 291101, 293201, 337039, 340577, 352831
Offset: 1
Keywords
Examples
a(1) = 3 since 3 is prime with 3-1, 3+1, prime(3)-1 = 4 and prime(3)+1 = 6 all practical. a(3) = 31 since 31 is prime with 31-1, 31+1, prime(31)-1 = 126 and prime(31)+1 = 128 all practical.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..2000
- Zhi-Wei Sun, Sandwiches with primes and practical numbers, a message to Number Theory List, Jan. 13, 2013.
- Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.
Programs
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Mathematica
f[n_]:=FactorInteger[n] Pow[n_,i_]:=Part[Part[f[n],i],1]^(Part[Part[f[n],i],2]) 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_]:=n>0&&(n<3||Mod[n,2]+Con[n]==0) n=0;Do[If[pr[Prime[k]-1]&&pr[Prime[k]+1]&&pr[Prime[Prime[k]]-1]&&pr[Prime[Prime[k]]+1],n=n+1;Print[n," ",Prime[k]]],{k,1,30201}]
Comments