A232861 Numbers k with k - 1, k + 1, prime(k) - k, prime(k) + k, k*prime(k) - 1, k*prime(k) + 1 all prime.

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

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