A186285 - OEIS

%I #44 Feb 20 2020 02:16:07

%S 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 83,89,97,101,103,107,109,113,125,127,131,137,139,149,151,157,163,167,

%U 173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271

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

%C Every positive rational number (in reduced form) is a product of a unique subset of these numbers and their multiplicative inverses.

%C 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 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 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 Amiram Eldar, <a href="/A186285/b186285.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 Prime powers which are not terms of this sequence:

%e   4 = 2^2 = 2^(3-1), 9 = 3^2 = 3^(3-1), 16 = 2^4 = 2^(3+1),

%e   25 = 5^2 = 5^(3-1), 32 = 2^5 = 2^(9-3-1), 49 = 7^2 = 7^(3-1),

%e   64 = 2^6 = 2^(9-3), 81 = 3^4 = 3^(3+1), 121 = 11^2 = 11^(3-1),

%e   128 = 2^7 = 2^(9-3+1).

%e "Factorization" of positive rational numbers into terms of this sequence:

%e (the balanced ternary digits {-1,0,+1} are represented here as {-,0,+})

%e          Factors from A186285          Balanced ternary representation

%e   33/14  = 11*(1/7)*3*(1/2)                      +0-0+-

%e   5/9    = (1/27)*5*3                            -0000000++0

%e   7/32   = (1/512)*8*7*2                         -...(96 0's)...++00+

%e   4/105  = 8*(1/7)*(1/5)*(1/3)*(1/2)             +----

%e   32     = 512*(1/8)*(1/2)                       +...(96 0's)...-000-

%e   81     = 27*3                                  +00000000+0

%t 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 (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 Cf. A050376, A186286, A186287, A185169.

%K nonn

%O 1,1

%A _Daniel Forgues_, Feb 17 2011