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A261281 Least positive integer k with prime(k)^2-2 and prime(prime(k))^2-2 both prime such that prime(k*n)^2-2 and prime(prime(k*n))^2-2 are all prime. 6
1, 1, 319, 134, 34, 62, 2, 536, 5215, 15, 3965, 2168, 34, 1, 1, 737, 2, 7075, 3699, 419, 132, 372, 14, 2, 34, 2, 52, 1, 668, 36561, 2, 48, 1239, 1, 401, 1613, 1646, 2472, 43, 31361, 134, 1103, 1, 5374, 6201, 466, 1, 1, 2118, 2, 1646, 1, 1343, 856, 28, 1868, 10324, 360, 2845, 6571, 65, 1, 419, 43, 1, 2, 2, 1, 889, 202 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Conjecture: a(n) exists for any n > 0. In general, any positive rational number r can be written as m/n with m and n in the set {k>0: prime(k)^2-2 and prime(prime(k))^2-2 are both prime}.

This implies that the sequence A237414 has infinitely many terms.

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..2000

Zhi-Wei Sun, Checking the conjecture for r = a/b with a,b = 1..300

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

EXAMPLE

a(2) = 1 since prime(1)^2-2 = 2^2-2 = 2, prime(prime(1))^2-2 = prime(2)^2-2 = 3^2-2 = 7, prime(1*2)^2-2 = 3^2-2 = 7, and prime(prime(1*2))^2-2 = prime(3)^2-2 = 5^2-2 = 23 are all prime.

a(3) = 319 since prime(319)^2-2 = 2113^2-2 = 4464767, prime(prime(319))^2-2 = prime(2113)^2-2 = 18443^2-2 = 340144247, prime(319*3)^2-2 = 7547^2-2 = 56957207, and prime(prime(3*319))^2-2 = prime(7547)^2-2 = 76757^2-2 = 5891637047 are all prime.

MATHEMATICA

f[n_]:=Prime[n]

q[n_]:=PrimeQ[f[n]^2-2]&&PrimeQ[f[f[n]]^2-2]

Do[k=0; Label[bb]; k=k+1; If[q[k]&&q[k*n], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 70}]

PROG

(PARI) a(n) = my(k=1); while (!isprime(prime(k)^2-2) || !isprime(prime(prime(k))^2-2) || !isprime(prime(k*n)^2-2) || !isprime(prime(prime(k*n))^2-2), k++); k; \\ Michel Marcus, Aug 14 2015

CROSSREFS

Cf. A000040, A049002, A062326, A237413, A237414, A259487.

Sequence in context: A281078 A262846 A225579 * A126838 A153048 A053020

Adjacent sequences:  A261278 A261279 A261280 * A261282 A261283 A261284

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Aug 14 2015

STATUS

approved

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Last modified October 22 10:39 EDT 2021. Contains 348160 sequences. (Running on oeis4.)