

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
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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

Jingrun 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), 157176.
ZhiWei 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 ChinaJapan Seminar (Fukuoka, Oct. 28  Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169187.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..2000
ZhiWei Sun, Checking the conjecture for r = a/b with a,b = 1..100
ZhiWei 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

ZhiWei Sun, Aug 14 2015


STATUS

approved



