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A186285 Numbers of the form p^(3^k) where p is prime and k >= 0.

%I A186285 #44 Feb 20 2020 02:16:07
%S A186285 2,3,5,7,8,11,13,17,19,23,27,29,31,37,41,43,47,53,59,61,67,71,73,79,
%T A186285 83,89,97,101,103,107,109,113,125,127,131,137,139,149,151,157,163,167,
%U A186285 173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271
%N A186285 Numbers of the form p^(3^k) where p is prime and k >= 0.
%C A186285 Every positive rational number (in reduced form) is a product of a unique subset of these numbers and their multiplicative inverses.
%C A186285 The factorization of positive rational numbers into prime powers of the form p^(3^k), k >= 0, (A186285) and their multiplicative inverses, allows each of those prime powers and their multiplicative inverses to be used at most once, since this corresponds to the balanced ternary representation of the exponents of the prime powers p^a and their multiplicative inverses of the prime factorization of positive rational numbers.
%C A186285 These are to the "Fermi-Dirac primes" [as _Daniel Forgues_ has denoted numbers of the form p^(2^k) where p is prime and k >= 0 (A050376)] as 3 is to 2. - _Jonathan Vos Post_, Mar 16 2013
%C A186285 Every integer > 1 is a unique product of terms of this sequence with no more than 2 the same. For example, 104 = 8*13, 27 = 27, 32 = 2^2*8. - _Vladimir Shevelev_, Sep 09 2013
%H A186285 Amiram Eldar, <a href="/A186285/b186285.txt">Table of n, a(n) for n = 1..10000</a>
%H A186285 S. Litsyn and V. Shevelev, <a href="http://www.emis.de/journals/INTEGERS/papers/h33/h33.Abstract.html">On Factorization of Integers with Restrictions on the Exponents</a>, INTEGERS: The Electronic Journal of Combinatorial Number Theory 7(2007), #A33 (case k=2).
%e A186285 Prime powers which are not terms of this sequence:
%e A186285   4 = 2^2 = 2^(3-1), 9 = 3^2 = 3^(3-1), 16 = 2^4 = 2^(3+1),
%e A186285   25 = 5^2 = 5^(3-1), 32 = 2^5 = 2^(9-3-1), 49 = 7^2 = 7^(3-1),
%e A186285   64 = 2^6 = 2^(9-3), 81 = 3^4 = 3^(3+1), 121 = 11^2 = 11^(3-1),
%e A186285   128 = 2^7 = 2^(9-3+1).
%e A186285 "Factorization" of positive rational numbers into terms of this sequence:
%e A186285 (the balanced ternary digits {-1,0,+1} are represented here as {-,0,+})
%e A186285          Factors from A186285          Balanced ternary representation
%e A186285   33/14  = 11*(1/7)*3*(1/2)                      +0-0+-
%e A186285   5/9    = (1/27)*5*3                            -0000000++0
%e A186285   7/32   = (1/512)*8*7*2                         -...(96 0's)...++00+
%e A186285   4/105  = 8*(1/7)*(1/5)*(1/3)*(1/2)             +----
%e A186285   32     = 512*(1/8)*(1/2)                       +...(96 0's)...-000-
%e A186285   81     = 27*3                                  +00000000+0
%t A186285 ofTheFormQ[n_] := PrimeQ[n] || Length[fi = FactorInteger[n]] == 1 && IntegerQ[Log[3, fi[[1, 2]]]]; Select[Range[2, 300], ofTheFormQ] (* _Jean-François Alcover_, Sep 09 2013 *)
%o A186285 (PARI) lista(nn) = {for (i=1, nn, if (isprime(i), print1(i, ", "), if ((pow = ispower(i, ,&p)) && isprime(p), if ((pow == 3) || ((ispower(pow, ,&k) && (k==3))), print1(i, ", ");););););} \\ _Michel Marcus_, Jun 12 2013
%Y A186285 Cf. A050376, A186286, A186287, A185169.
%K A186285 nonn
%O A186285 1,1
%A A186285 _Daniel Forgues_, Feb 17 2011