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

2, 3, 5, 7, 8, 11, 13, 17, 19, 23, 27, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271

Offset: 1

Author

Daniel Forgues, Feb 17 2011

Keywords

• nonn

Comments

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

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.

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

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

Examples

			Prime powers which are not terms of this sequence:
  4 = 2^2 = 2^(3-1), 9 = 3^2 = 3^(3-1), 16 = 2^4 = 2^(3+1),
  25 = 5^2 = 5^(3-1), 32 = 2^5 = 2^(9-3-1), 49 = 7^2 = 7^(3-1),
  64 = 2^6 = 2^(9-3), 81 = 3^4 = 3^(3+1), 121 = 11^2 = 11^(3-1),
  128 = 2^7 = 2^(9-3+1).
"Factorization" of positive rational numbers into terms of this sequence:
(the balanced ternary digits {-1,0,+1} are represented here as {-,0,+})
         Factors from A186285          Balanced ternary representation
  33/14  = 11*(1/7)*3*(1/2)                      +0-0+-
  5/9    = (1/27)*5*3                            -0000000++0
  7/32   = (1/512)*8*7*2                         -...(96 0's)...++00+
  4/105  = 8*(1/7)*(1/5)*(1/3)*(1/2)             +----
  32     = 512*(1/8)*(1/2)                       +...(96 0's)...-000-
  81     = 27*3                                  +00000000+0
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• S. Litsyn and V. Shevelev, On Factorization of Integers with Restrictions on the Exponents, INTEGERS: The Electronic Journal of Combinatorial Number Theory 7(2007), #A33 (case k=2).

Crossrefs

Cf. A050376, A186286, A186287, A185169.

Programs

Mathematica

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 *)

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