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A261352
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Primes p such that prime(p)+2 = prime(q)*prime(r) for distinct primes q and r.
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5
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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
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1,1
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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