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A178700
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Lonely primes between two consecutive nontrivial powers.
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0
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OFFSET
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1,1
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COMMENTS
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While the number of perfect powers < n is ~ sqrt(n) and the number of primes < n is ~ n/log(n), this does not preclude more terms from existing, but it does make it very unlikely. [Robert G. Wilson v, Jun 10 2010]
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LINKS
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EXAMPLE
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5^3 < 127 < 2^7;
6^3 < 223 < 15^2;
3^5 < 251 < 2^8;
40^3 < 64007 < 253^2;
109^3 < 1295033 < 1138^2.
50354^2 < 2535525373 < 76^5;
8158^3 < 542939080319 < 736844^2. (End)
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MATHEMATICA
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nextPerfectPower[n_] := Block[{k = n + 1}, While[ GCD @@ Last /@ FactorInteger@k == 1, k++ ]; k]; a = 1; b = 3; lst = {}; While[a < 15 10^8, If[ PrimePi@b == 1 + PrimePi@a, p = NextPrime@a; AppendTo[lst, p]; Print@p]; a = b; b = nextPerfectPower@b]; lst (* Robert G. Wilson v, Jun 10 2010 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Marot Alain (marot.alain(AT)orange.fr), Jun 05 2010
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EXTENSIONS
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STATUS
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approved
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