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 A232861 Numbers k with k - 1, k + 1, prime(k) - k, prime(k) + k, k*prime(k) - 1, k*prime(k) + 1 all prime. 8
 22110, 23742, 128238, 275592, 346560, 1061910, 1281522, 1339002, 1378188, 1461600, 1850130, 2064150, 2354952, 2478270, 2523708, 2689260, 2694300, 3916638, 4422618, 4933530, 6179082, 6541080, 6641562, 6740478, 6759030, 7315812, 8484798, 8711010, 9133308, 9687720 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Obviously, each term of the sequence is a multiple of 6. Conjecture: (i) This sequence contains infinitely many terms. (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. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..2000 Z.-W. Sun, On a^n+ bn modulo m, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014. Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2017. EXAMPLE 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. MATHEMATICA n=0 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}] CROSSREFS Cf. A000040, A014574, A014689, A064269, A064270, A064370, A105645, A232443, A232463, A232548. Sequence in context: A049535 A091459 A224574 * A250672 A062564 A043590 Adjacent sequences: A232858 A232859 A232860 * A232862 A232863 A232864 KEYWORD nonn AUTHOR Zhi-Wei Sun, Dec 01 2013 STATUS approved

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Last modified February 21 22:50 EST 2024. Contains 370239 sequences. (Running on oeis4.)