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%I #12 Jul 13 2015 04:27:14
%S 4,522,1932,5100,6132,6552,8220,18312,18540,22110,29568,45342,70488,
%T 70950,92220,105360,109662,114600,116532,117192,123552,128982,131838,
%U 132762,136710,148302,149160,166848,177012,183438,197340,206280,233550,235008,257868,272808,273900,276780,279708,286590
%N Practical numbers m with m-1 and m+1 both prime, and prime(m)-1 and prime(m)+1 both practical.
%C Conjecture: The sequence contains infinitely many terms. In other words, there are infinitely many positive integers n such that {prime(n)-1, prime(n), prime(n)+1} is a "sandwich of the first kind" (A210479) and {n-1, n, n+1} is a "sandwich of the second kind" (A258838).
%C This implies that there are infinitely many sandwiches of the first kind and also there are infinitely many sandwiches of the second kind.
%H Zhi-Wei Sun, <a href="/A257922/b257922.txt">Table of n, a(n) for n = 1..10000</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) = 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.
%e 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.
%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[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}]
%Y Cf. A000040, A005153, A014574, A209236, A210479, A257924, A258836, A258838, A259539.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Jul 12 2015
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