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A261462
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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.
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2
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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
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OFFSET
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1,2
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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