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A259628
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Numbers m with m-1, m+1, prime(m)+2, prime(m)-m, prime(m)+m, m*prime(m)-1 and m*prime(m)+1 all prime.
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1
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2523708, 6740478, 6759030, 14655522, 22885698, 28384200, 44630148, 71742300, 87002328, 87466500, 89842200, 153110622, 153647490, 184373490, 283232040, 312124920, 366318960, 408770670, 412216920, 439429128, 456486030, 486730398, 517602600, 606159978, 607942848, 675661080, 855983352, 869593998, 923864562, 971400672
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OFFSET
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1,1
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COMMENTS
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Conjecture: The sequence contains infinitely many terms.
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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) = 2523708 since the seven numbers 2523707, 2523709, prime(2523708)+2 = 41578739+2 = 41578741, prime(2523708)-2523708 = 41578739-2523708 = 39055031, prime(2523708)+2523708 = 41578739+2523708 = 44102447, 2523708*prime(2523708)-1 = 2523708*41578739-1 = 104932596244211 and 2523708*prime(2523708)+1 = 2523708*41578739+1 = 104932596244213 are all prime.
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MATHEMATICA
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TW[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]
n=0; Do[If[PrimeQ[Prime[k]+2]&&PrimeQ[Prime[Prime[k]+1]+2]&&PrimeQ[Prime[Prime[k]+1]-Prime[k]-1]&&PrimeQ[Prime[Prime[k]+1]+Prime[k]+1]&&TW[(Prime[k]+1)Prime[Prime[k]+1]], n=n+1; Print[n, " ", Prime[k]+1]], {k, 1, 5*10^7}]
allprQ[n_]:=Module[{p=Prime[n]}, AllTrue[{n-1, n+1, p+2, p-n, p+n, n*p-1, n*p+1}, PrimeQ]]; Select[Range[98*10^7], allprQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 26 2016 *)
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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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