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A261353
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Least positive integer k such that prime(prime(k))*prime(prime(k*n)) = prime(p)+2 for some prime p.
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4
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11, 2, 1, 606, 350, 166, 53, 1865, 7, 45, 1308, 68, 215, 61, 256, 13, 248, 90, 1, 1779, 796, 1, 4, 444, 650, 55, 157, 303, 82, 84, 25, 3, 1912, 621, 128, 205, 164, 1091, 61, 12, 337, 1, 303, 15, 23, 418, 212, 23, 2494, 1, 472, 771, 1, 36, 8, 46, 8, 18, 264, 22, 725, 85, 65, 231, 606, 3, 1, 43, 144, 164
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OFFSET
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1,1
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COMMENTS
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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(prime(m))*prime(prime(n)) = prime(p)+2 for some prime p.
This implies that the sequence A261352 has infinitely many terms.
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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 prime(prime(11))*prime(prime(11*1)) = prime(31)^2 = 127^2 = 16129 = prime(1877)+2 with 1877 prime.
a(4) = 606 since prime(prime(606))*prime(prime(606*4)) = prime(4457)*prime(21589) = 42643*244471 = 10424976853 = prime(473490161)+2 with 473490161 prime.
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MATHEMATICA
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f[n_]:=Prime[Prime[n]]
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
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}]
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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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