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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Dec 01 2013
STATUS
approved