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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
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 (list; graph; refs; listen; history; text; internal format)
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 * A256089 A252008 A252015

Adjacent sequences:  A261580 A261581 A261582 * A261584 A261585 A261586

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Aug 25 2015

STATUS

approved

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Last modified January 23 01:30 EST 2020. Contains 331166 sequences. (Running on oeis4.)