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A268043
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Numbers k such that k^3 - 1 and k^3 + 1 are both semiprimes.
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8
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6, 1092, 1932, 2730, 4158, 6552, 11172, 25998, 30492, 55440, 76650, 79632, 85092, 102102, 150990, 152082, 152418, 166782, 211218, 235662, 236208, 248640, 264600, 298410, 300300, 301182, 317772, 380310, 387198, 441798, 476028, 488418
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OFFSET
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1,1
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COMMENTS
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Obviously, k+1 and k-1 are always prime numbers.
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LINKS
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EXAMPLE
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a(1) = 6 because 6^3-1 = 215 = 5*43 and 6^3+1 = 217 = 7*31, therefore 215, 217 are both semiprimes.
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MATHEMATICA
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Select[Range[500000], PrimeOmega[#^3 - 1] == PrimeOmega[#^3 + 1] == 2 &]
Select[Range[10^6], And @@ PrimeQ[{# - 1, # + 1, #^2 - # + 1, #^2 + # + 1}] &] (* Amiram Eldar, Apr 19 2024 *)
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PROG
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(Magma) IsSemiprime:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [2..300000] | IsSemiprime(n^3+1) and IsSemiprime(n^3-1) ];
(PARI) isok(n) = (bigomega(n^3-1) == 2) && (bigomega(n^3+1) == 2); \\ Michel Marcus, Jan 26 2016
(PARI) is(n) = isprime(n - 1) && isprime(n + 1) && isprime(n^2 - n + 1) && isprime(n^2 + n + 1); \\ Amiram Eldar, Apr 19 2024
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CROSSREFS
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KEYWORD
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nonn,easy,changed
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AUTHOR
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STATUS
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approved
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