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A261282 Least positive integer k such that prime(k)*prime(k*n) = prime(p)+2 for some prime p. 4
14, 60, 135, 41, 199, 2, 2, 2, 61, 2, 183, 25, 15, 12, 47, 143, 110, 294, 117, 88, 22, 402, 26, 269, 116, 145, 164, 6, 10, 488, 2, 44, 120, 4, 127, 144, 119, 704, 1058, 368, 104, 2, 6, 214, 4, 129, 2, 3, 301, 2, 2, 466, 20, 107, 280, 14, 337, 12, 22, 12, 242, 1705, 415, 10, 115, 50, 2, 420, 4, 15 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Conjecture: a(n) exists for any n > 0. In general, any positive rational number r can be written as m/n, where m and n are positive integers such that prime(m)*prime(n) = prime(p)+2 for some prime p.

For example, 14/19 = 24528/33288, and prime(24528)*prime(33288) = 281153*392723 = 110415249619 = prime(4528436431)+2 with 4528436431 prime.

The conjecture implies that there are infinitely many primes p such that prime(p)+2 is a product of two primes. Recall that a prime p is called a Chen prime if p+2 is a product of at most two primes.

REFERENCES

Jing-run Chen, On the representation of a large even integer as the sum of a prime and a product of at most two primes, Sci. Sinica 16(1973), 157-176.

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..100

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

EXAMPLE

a(2) = 60 since prime(60)*prime(60*2) = 281*659 = 185179 = prime(16763)+2 with 16763 prime.

MATHEMATICA

f[n_]:=Prime[n]

PQ[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]

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, 70}]

CROSSREFS

Cf. A000040, A109611, A257926, A259487, A261281.

Sequence in context: A120371 A062022 A277986 * A158058 A100171 A063492

Adjacent sequences:  A261279 A261280 A261281 * A261283 A261284 A261285

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Aug 14 2015

STATUS

approved

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Last modified October 22 05:06 EDT 2018. Contains 316432 sequences. (Running on oeis4.)