A257922 Practical numbers m with m-1 and m+1 both prime, and prime(m)-1 and prime(m)+1 both practical.
4, 522, 1932, 5100, 6132, 6552, 8220, 18312, 18540, 22110, 29568, 45342, 70488, 70950, 92220, 105360, 109662, 114600, 116532, 117192, 123552, 128982, 131838, 132762, 136710, 148302, 149160, 166848, 177012, 183438, 197340, 206280, 233550, 235008, 257868, 272808, 273900, 276780, 279708, 286590
Offset: 1
Keywords
Examples
a(1) = 4 since 4 is paractical with 4-1 and 4+1 twin prime, and prime(4)-1 = 6 and prime(4)+1 = 8 are both practical. a(2) = 522 since 522 is paractical with 522-1 and 522+1 twin prime, and prime(522)-1 = 3738 and prime(522)+1 = 3740 are both practical.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
- 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[PrimeQ[Prime[k]+2]&&pr[Prime[k]+1]&&pr[Prime[Prime[k]+1]-1]&&pr[Prime[Prime[k]+1]+1],n=n+1;Print[n," ",Prime[k]+1]],{k,1,24962}]
Comments