%I #11 Jul 13 2015 04:27:25

%S 3,7,31,89,199,8009,11551,20129,23549,38609,47501,67231,96221,97001,

%T 103409,111871,120473,131071,143261,146681,168869,174761,183091,

%U 193951,196181,208279,208961,219727,229769,237691,238519,240641,247759,270271,290249,291101,293201,337039,340577,352831

%N Primes p with p-1, p+1, prime(p)-1 and prime(p)+1 all practical.

%C Conjecture: The sequence contains infinitely many terms. In other words, there are infinitely many prime numbers p such that {p-1, p, p+1} and {prime(p)-1, prime(p), prime(p)+1} are both "sandwiches of the first kind" (A210479).

%H Zhi-Wei Sun, <a href="/A257924/b257924.txt">Table of n, a(n) for n = 1..2000</a>

%H Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;20e70044.1301">Sandwiches with primes and practical numbers</a>, a message to Number Theory List, Jan. 13, 2013.

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588.

%e a(1) = 3 since 3 is prime with 3-1, 3+1, prime(3)-1 = 4 and prime(3)+1 = 6 all practical.

%e a(3) = 31 since 31 is prime with 31-1, 31+1, prime(31)-1 = 126 and prime(31)+1 = 128 all practical.

%t f[n_]:=FactorInteger[n]

%t Pow[n_,i_]:=Part[Part[f[n],i],1]^(Part[Part[f[n],i],2])

%t 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 pr[n_]:=n>0&&(n<3||Mod[n,2]+Con[n]==0)

%t 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}]

%Y Cf. A000040, A005153, A209236, A210479, A257922, A258838, A259539.

%K nonn

%O 1,1

%A _Zhi-Wei Sun_, Jul 13 2015