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%I #32 Aug 05 2019 09:27:12
%S 22110,23742,128238,275592,346560,1061910,1281522,1339002,1378188,
%T 1461600,1850130,2064150,2354952,2478270,2523708,2689260,2694300,
%U 3916638,4422618,4933530,6179082,6541080,6641562,6740478,6759030,7315812,8484798,8711010,9133308,9687720
%N Numbers k with k - 1, k + 1, prime(k) - k, prime(k) + k, k*prime(k) - 1, k*prime(k) + 1 all prime.
%C Obviously, each term of the sequence is a multiple of 6.
%C Conjecture: (i) This sequence contains infinitely many terms.
%C (ii) Let P(x) be a non-constant integer-valued polynomial with positive leading coefficient. Then, there are infinitely many positive integers k with prime(k) - k in the range P(Z) = {P(m): m is an integer}, if and only if the degree of P(x) is at most 3. We may also replace prime(k) - k by prime(k) + k.
%H Zhi-Wei Sun, <a href="/A232861/b232861.txt">Table of n, a(n) for n = 1..2000</a>
%H Z.-W. Sun, <a href="http://arxiv.org/abs/1312.1166">On a^n+ bn modulo m</a>, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.
%H Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014-2017.
%e a(1) = 22110 with the six numbers 22110 - 1 = 22109, 22110 + 1 = 22111, prime(22110) - 22110 = 228841, prime(22110) + 22110 = 273061, 22110*prime(22110) - 1 = 5548526609, 22110*prime(22110) + 1 = 5548526611 all prime.
%t n=0
%t Do[If[PrimeQ[k-1]&&PrimeQ[k+1]&&PrimeQ[Prime[k]-k]&& PrimeQ[Prime[k]+k]&& PrimeQ[k*Prime[k]-1]&& PrimeQ[k*Prime[k]+1],n=n+1;Print[n," ",k]],{k,1,9700000}]
%Y Cf. A000040, A014574, A014689, A064269, A064270, A064370, A105645, A232443, A232463, A232548.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Dec 01 2013
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