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A132282
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Near-cube primes: primes of the form p^3 + 2, where p is noncomposite.
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4
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2, 3, 29, 127, 24391, 357913, 571789, 1442899, 5177719, 18191449, 30080233, 73560061, 80062993, 118370773, 127263529, 131872231, 318611989, 344472103, 440711083, 461889919, 590589721, 756058033, 865523179, 1095912793
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OFFSET
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1,1
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COMMENTS
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The corresponding near-cube prime indices q are A132281. Analog of near-square primes. After a(1) = 2, all values must be odd. Numbers of the form n^2+2 for n=1, 2, ... are 3, 6, 11, 18, 27, 38, 51, 66, 83, 102, ... (A059100). These are prime for indices n = 1, 3, 9, 15, 21, 33, 39, 45, 57, 81, 99, ... (A067201), corresponding to the near-square primes 3, 11, 83, 227, 443, 1091, 1523, 2027, ... (A056899). Helfgott proves with minor conditions that: "Let f be a cubic polynomial. Then there are infinitely many primes p such that f(p) is squarefree." Note that 47^3 + 2 = 103825 = 5^2 * 4153 and similarly 97^3 + 2 is divisible by 5^2, but otherwise an infinite number of p^3+2 are squarefree.
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LINKS
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FORMULA
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EXAMPLE
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a(1) = 0^3 + 2 = 2 is prime and 0 is noncomposite.
a(2) = 1^3 + 2 = 3 is prime and 1 is noncomposite.
a(3) = 3^3 + 2 = 29 is prime and 3 is prime.
a(4) = 5^3 + 2 = 127 is prime and 5 is prime.
a(5) = 29^3 + 2 = 24391 is prime and 29 is prime.
45^3 + 2 = 91127 is prime, but not in this sequence because 45 is not prime.
63^3 + 2 = 250049 is prime, but not in this sequence because 63 is not prime.
a(6) = 71^3 + 2 = 357913 is prime.
a(7) = 83^3 + 2 = 571789 is prime.
a(8) = 113^3 + 2 = 1442899 is prime.
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MATHEMATICA
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Join[{2, 5}, Select[Prime[Range[200]]^3 + 2, PrimeQ[ # ] &]] (* Stefan Steinerberger, Aug 17 2007 *)
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PROG
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(PARI) v=[2, 3]; forprime(p=3, 1e4, if(isprime(t=p^3+2), v=concat(v, t))); t \\ Charles R Greathouse IV, Feb 14 2011
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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