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
Keywords
Examples
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.
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.
Programs
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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}]
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