A261583 Least positive integer k such that prime(prime(prime(k)))+ prime(prime(prime(k*n))) = 2*prime(prime(p)) for some prime p.

1, 755, 4648, 1335, 1096, 14708, 5964, 636, 1063, 13019, 9808, 2776, 2580, 2797, 6421, 1573, 2432, 4790, 862, 1855, 566, 2145, 18554, 35634, 5264, 1293, 39402, 1445, 2397, 17930, 586, 2526, 24571, 18403, 5480, 366, 5159, 9710, 179, 4469, 6757, 7866, 263, 1701, 2941, 477, 5032, 10705, 3494, 8597, 953, 11954, 2586, 689, 9456, 1309, 8651, 12538, 4106, 13762

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(prime(m))) + prime(prime(prime(n))))/2 = prime(prime(p)) for some prime p.

(ii) Let p(1,n) = prime(n), and p(m+1,n) = p(m,prime(n)) for m,n = 1,2,3,.... Then, for any integers m > 0 and k > 2, the sequence p(m,n) (n = 1,2,3,...) contains infinitely many nontrivial k-term arithmetic progressions.

(iii) Let m be any positive integer. Then the sequence p(m,n)^(1/n) (n = 1,2,3,...) is strictly decreasing. Also, for any relatively prime integers q > 0 and r, there are infinitely many n > 0 such that p(m,n) == r (mod q).

Note that part (ii) of the conjecture extends the Green-Tao theorem and the third part of the conjecture in A261462. Also, part (iii) in the case m = 1 reduces to the Firoozbakht conjecture on primes and Dirichlet's theorem on primes in 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..200

Zhi-Wei Sun, Checking part (i) of the conjecture for r = a/b with a,b = 1..100

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

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

Example

a(2) = 755 since prime(prime(prime(755))) + prime(prime(prime(755*2))) = prime(prime(5741)) + prime(prime(12641)) = prime(56611) + prime(135671) = 700897 + 1808581 = 2*1254739 = 2*prime(96797) = 2*prime(prime(9319)) with 9319 prime.

Mathematica

f[n_]:=Prime[Prime[Prime[n]]]
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]&&PrimeQ[PrimePi[PrimePi[p]]]
Do[k=0; Label[bb]; k=k+1; If[PQ[(f[k]+f[k*n])/2], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 60}]

Crossrefs

Cf. A000040, A006450, A261437, A261462.

Sequence in context: A252917 A252918 A232341 * A332879 A256089 A252008

Adjacent sequences: A261580 A261581 A261582 * A261584 A261585 A261586

Keyword

nonn

Author

Zhi-Wei Sun, Aug 25 2015

Status

approved