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A261528
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Least positive integer k such that both k and k*n belong to the set {m>0: prime(m)+2 is prime with prime(prime(m)+2) = prime(prime(m))+6}.
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2
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2, 891, 81002, 814812, 86050, 5917, 65527, 109853, 2563344, 25379, 2640232, 266076, 775889, 67387, 68111, 37950, 353416, 347139, 56390, 11299, 89491, 545458, 910786, 353416, 1913477, 9025, 111569, 511796, 1456228, 37909, 1494675, 212092, 69352, 107769, 300657, 1155675, 391972, 1073031, 55074, 49892
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OFFSET
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1,1
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COMMENTS
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Conjecture: Any positive rational number r can be written as m/n with m and n in the set {k>0: prime(k)+2 is prime with prime(prime(k)+2) = prime(prime(k))+6}.
This implies that there are infinitely many twin prime pairs {p, p+2} with prime(p+2) - prime(p) = 6.
Note that if prime(n+2)-prime(n) = 6 then prime(n+1)-prime(n) = 2 or 4.
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REFERENCES
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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) = 2 since 2*1 = 2, and prime(2)+2 = 3+2 = 5 is prime with prime(5)-prime(3) = 11-5 = 6.
a(2) = 891 since prime(891)+2 = 6947 + 2 = 6949 is prime with prime(6949)-prime(6947) = 70123-70117 = 6, and prime(891*2)+2 = 15269 + 2 = 15271 is prime with prime(15271)-prime(15269) = 167119-167113 = 6.
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MATHEMATICA
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f[n_]:=Prime[n]
PQ[k_]:=PrimeQ[f[k]+2]&&f[f[k]+2]-f[f[k]]==6
Do[k=0; Label[bb]; k=k+1; If[PQ[k]&&PQ[k*n], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 40}]
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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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