

A261352


Primes p such that prime(p)+2 = prime(q)*prime(r) for distinct primes q and r.


5



11, 23, 167, 197, 223, 317, 359, 461, 593, 619, 859, 1283, 1289, 1327, 1487, 1759, 1879, 2557, 2579, 2749, 2879, 3617, 4159, 4783, 5081, 5333, 5531, 5689, 5783, 5867, 6427, 6521, 7589, 7681, 7727, 7753, 9041, 9157, 9283, 9479, 10111, 10289, 10853, 11261, 11779, 11867, 12541, 13309, 13399, 13687
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OFFSET

1,1


COMMENTS

Conjecture: The sequence has infinitely many terms.
See also A261354 for a similar conjecture, and A261353 for a stronger conjecture.
Recall that a prime p is called a Chen prime if p+2 is a product of at most two primes. It is known that there are infinitely many Chen 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..10000
ZhiWei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.


EXAMPLE

a(1) = 11 since 11 is a prime, and prime(11)+2 = 3*11 = prime(2)*prime(5) with 2 and 5 both prime.
a(2) = 23 since 23 is a prime, and prime(23)+2 = 5*17 = prime(3)*prime(7) with 3 and 7 both prime.


MATHEMATICA

Dv[n_]:=Divisors[n]
PQ[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]
q[n_]:=Length[Dv[n]]==4&&PQ[Part[Dv[n], 2]]&&PQ[Part[Dv[n], 3]]
f[k_]:=Prime[Prime[k]]+2
n=0; Do[If[q[f[k]], n=n+1; Print[n, " ", Prime[k]]], {k, 1, 1620}]


CROSSREFS

Cf. A000040, A109611, A261282, A261353, A261354, A261361.
Sequence in context: A106388 A171068 A091465 * A018979 A181147 A059327
Adjacent sequences: A261349 A261350 A261351 * A261353 A261354 A261355


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Aug 15 2015


STATUS

approved



