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%I #14 Dec 17 2017 10:07:18
%S 4,6,12,18,30,42,60,72,108,150,180,192,198,228,240,270,312,348,420,
%T 432,462,522,570,600,660,810,828,858,882,1020,1032,1050,1092,1152,
%U 1230,1290,1302,1320,1428,1452,1482,1488,1620,1722,1872,1932,1950,1998,2028,2088,2112,2130,2142,2268,2310,2340,2550,2592,2688,2730
%N Practical numbers q with q-1 and q+1 twin primes: "Sandwiches of the second kind".
%C The author introduced two kinds of "sandwiches" in 2013. The conjecture in A258836 essentially says that {a(m)/a(n): m,n = 1,2,3,...} coincides with the set of all positive rational numbers. This implies that the sequence contains infinitely many terms.
%H Zhi-Wei Sun, <a href="/A258838/b258838.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://maths.nju.edu.cn/~zwsun/176r.pdf">Conjectures on representations involving primes</a>, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer Proc. in Math. & Stat., Vol. 220, Springer, New York, 2017, pp. 279-310. (See also <a href="http://arxiv.org/abs/1211.1588">arXiv:1211.1588 [math.NT]</a>, 2012-2017.)
%e a(1) = 4 since 4 is practical with 4-1 and 4+1 twin prime.
%e a(2) = 6 since 6 is practical with 6-1 and 6+1 twin prime.
%e a(3) = 12 since 12 is practical with 12-1 and 12+1 twin prime.
%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 SW[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]&&pr[n]
%t n=0;Do[If[SW[m],n=n+1;Print[n," ",m]],{m,1,2730}]
%Y Cf. A000040, A001359, A005153, A006512, A014574, A209236, A210452, A210479, A258836.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Jun 12 2015
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