A261462 Least positive integer k such that prime(prime(k)), prime(prime(k*n)), prime(p) and prime(q) form a 4-term arithmetic progression for some pair of primes p and q.

1, 5109, 879, 27956, 103840, 32205, 21404, 1800, 4241, 81794, 2355, 4352, 14974, 8552, 26159, 17621, 91986, 52574, 73764, 4699, 12546, 27347, 71148, 6819, 333, 38830, 28809, 2058, 24609, 84, 11478, 226251, 21383, 54, 2930, 36423, 2602, 22, 47668, 15594, 19, 56106, 41913, 72620, 211070, 9022, 2587, 10316, 74965, 26852

Offset

1,2

Comments

Conjecture: (i) Any positive rational number r can be written as m/n, where m and n are positive integers such that (prime(prime(m))+prime(prime(n)))/2 = prime(p) for some prime p.

(ii) Any positive rational number r <= 1 can be written as m/n, where m and n are positive integers such that prime(prime(m)),prime(prime(n)), prime(p) and prime(q) form a 4-term arithmetic progression for some pair of primes p and q.

(iii) For any integer k > 2, the set {prime(p): p is prime} contains infinitely many nontrivial k-term arithmetic progressions.

References

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.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..100

B. Green and T. Tao, The primes contain arbitrary long arithmetic progressions, Annals of Math. 167(2008), 481-547.

Zhi-Wei Sun, Checking part (ii) of the conjecture for r = a/b with 1 <= a <= b <= 50

Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Example

a(2) = 5109 since prime(prime(5109)) = 608591, prime(prime(5109*2)) = 1401791, prime(162343) = 2194991, and prime(216023) = 2988191 form a 4-term arithmetic progression with 162343 and 216023 both prime.

Mathematica

f[n_]:=Prime[Prime[n]]
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
Do[k=0; Label[bb]; k=k+1; If[PQ[2*f[k*n]-f[k]]&&PQ[3*f[k*n]-2*f[k]], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 50}]

Crossrefs

Cf. A000040, A006450, A232502, A238289, A261385, A261395.

Sequence in context: A343987 A343986 A255209 * A215839 A258729 A259425

Adjacent sequences: A261459 A261460 A261461 * A261463 A261464 A261465

Keyword

nonn

Author

Zhi-Wei Sun, Aug 20 2015

Status

approved