Search: id:A132282
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%I A132282
%S A132282 2,5,29,127,24391,357913,571789,1442899,5177719,18191449,30080233,
%T A132282 73560061,80062993,118370773,127263529,131872231,318611989,344472103,
%U A132282 440711083,461889919,590589721,756058033,865523179,1095912793
%N A132282 Near-cube primes: primes of the form p^3 + 2, where p is noncomposite.
%C A132282 The corresponding near-cube prime indices q are A132281. Analogue of 
               near-square primes. After a(1) = 1, 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 square-free." 
               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 square-free.
%H A132282 Harald Andres Helfgott, <a href="http://arXiv.org/pdf/0706.1497">Power-free 
               values, repulsion between points, differing beliefs and the existence 
               of error</a>
%F A132282 a(n) = A132281(n)^3 + 2. {p in A000040 such that for some q = 0, 1, or 
               q in A000040, we have p = A067200(q) = A084380(q) = q^3 + 2 is in 
               A000040}.
%e A132282 a(1) = 0^3 + 2 = 2 is prime and 0 is noncomposite.
%e A132282 a(2) = 1^3 + 2 = 5 is prime and 1 is noncomposite.
%e A132282 a(3) = 3^3 + 2 = 29 is prime and 3 is prime.
%e A132282 a(4) = 5^3 + 2 = 127 is prime and 5 is prime.
%e A132282 a(5) = 29^3 + 2 = 24391 is prime and 29 is prime.
%e A132282 45^3 + 2 = 91127 is prime, but not in this sequence because 45 is not 
               prime.
%e A132282 63^3 + 2 = 250049 is prime, but not in this sequence because 63 is not 
               prime.
%e A132282 a(6) = 71^3 + 2 = 357913 is prime.
%e A132282 a(7) = 83^3 + 2 = 571789 is prime.
%e A132282 a(8) = 113^3 + 2 = 1442899 is prime.
%t A132282 Join[{2, 5}, Select[Prime[Range[200]]^3 + 2, PrimeQ[ # ] &]] - Stefan 
               Steinerberger (stefan.steinerberger(AT)gmail.com), Aug 17 2007
%Y A132282 Cf. A000040, A056899, A059100, A067200, A067201, A084380, A132281.
%Y A132282 Sequence in context: A057794 A073715 A104083 this_sequence A007014 A098682 
               A108367
%Y A132282 Adjacent sequences: A132279 A132280 A132281 this_sequence A132283 A132284 
               A132285
%K A132282 easy,nonn
%O A132282 1,1
%A A132282 Jonathan Vos Post (jvospost3(AT)gmail.com), Aug 16 2007
%E A132282 More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Aug 17 2007

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