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%I #8 Aug 24 2015 03:38:41
%S 3,358712,34772,79631,1822685,22865,2066,2593722,26,3418900,26,711611,
%T 286,1493190,882854,513312,1707237,788232,913695,1980985,7147,443152,
%U 479580,2589105,865432,265243,103641,160536,398360,851672
%N Least positive integer m such that both m and m*n belong to the set {k>0: prime(k)+2, prime(k)+6, prime(k)+8 are all prime}.
%C Conjecture: (i) Each positive rational number r can be written as m/n with m and n in the set {k>0: prime(k)+2, prime(k)+6 and prime(k)+8 are all prime}.
%C (ii) Any positive rational number r can be written as m/n with m and n in the set {k>0: prime(k)+4, prime(k)+6 and prime(k)+10 are all prime}.
%C For example, 3/4 = 20723892/27631856, and prime(20723892)+2 = 387875561+2 = 387875563, prime(20723892)+6 = 387875567, prime(20723892)+8 = 387875569, prime(27631856)+2 = 525608591+2 =525608593, prime(27631856)+6 = 525608597, prime(27631856)+8 = 525608599 are all prime. Also, 3/4 = 599478/799304, and prime(599478)+4 = 8951857+4 = 8951861, prime(599478)+6 = 8951863, prime(599478)+10 = 8951867, prime(799304)+4 = 12183943+4 = 12183947, prime(799304)+6 = 12183949, prime(799304)+10 = 12183953 are all prime.
%C Part (i) of the conjecture implies that there are infinitely many primes p with p+2, p+6 and p+8 all prime, while part (ii) implies that there are infinitely many primes p with p+4, p+6 and p+10 all prime.
%D Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.
%H Zhi-Wei Sun, <a href="/A261541/b261541.txt">Table of n, a(n) for n = 1..100</a>
%H Zhi-Wei Sun, <a href="/A261541/a261541.txt">Checking part (i) of the conjecture for r = a/b with a,b = 1..25</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.
%e a(1) = 3 since 3*1 = 3, and prime(3)+2 = 5+2 =7, prime(3)+6 = 11 and prime(3)+8 = 13 are all prime.
%e a(2) = 358712 since prime(358712)+2 = 5158031+2 = 5158033, prime(358712)+6 = 5158037, prime(358712)+8 = 5158039, prime(358712*2)+2 = 10852601+2 = 10852603, prime(358712*2)+6 = 10852607 and prime(358712*2)+8 = 10852609 are all prime.
%t f[n_]:=Prime[n]
%t PQ[k_]:=PrimeQ[f[k]+2]&&PrimeQ[f[k]+6]&&PrimeQ[f[k]+8]
%t Do[k=0;Label[bb];k=k+1;If[PQ[k]&&PQ[k*n],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,30}]
%Y Cf. A000040, A007530, A052378, A236511, A259487, A259540.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Aug 24 2015
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