A064614 Exchange 2 and 3 in the prime factorization of n.

1, 3, 2, 9, 5, 6, 7, 27, 4, 15, 11, 18, 13, 21, 10, 81, 17, 12, 19, 45, 14, 33, 23, 54, 25, 39, 8, 63, 29, 30, 31, 243, 22, 51, 35, 36, 37, 57, 26, 135, 41, 42, 43, 99, 20, 69, 47, 162, 49, 75, 34, 117, 53, 24, 55, 189, 38, 87, 59, 90, 61, 93, 28, 729, 65, 66, 67, 153, 46

Offset: 1

Author

Reinhard Zumkeller, Sep 25 2001

Keywords

• nonn
• mult
• nice
• easy

Comments

A self-inverse permutation of the natural numbers.

a(1) = 1, a(2) = 3, a(3) = 2, a(p) = p for primes p > 3 and a(u * v) = a(u) * a(v) for u, v > 0.

A permutation of the natural numbers: a(a(n)) = n for all n and a(n) = n iff n = 6^k * m for k >= 0 and m > 0 with gcd(m, 6) = 1 (see A064615).

A000244 and A000079 give record values and where they occur. - Reinhard Zumkeller, Feb 08 2010

Examples

			a(15) = a(3*5) = a(3)*a(5) = 2*5 = 10;
a(16) = a(2^4) = a(2)^4 = 3^4 = 81;
a(17) = 17;
a(18) = a(2*3^2) = a(2)*a(3^2) = 3*a(3)^2 = 3*2^2 = 12.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• A. B. Frizell, Certain non-enumerable sets of infinite permutations, Bull. Amer. Math. Soc. 21, No. 10 (1915), 495-499.
• Index entries for sequences that are permutations of the natural numbers.
• Index to divisibility sequences.

Crossrefs

Cf. A000079, A000244, A007814, A007949, A027746, A064615, A065330, A253046, A253047.

Programs

Haskell

a064614 1 = 1
a064614 n = product $ map f $ a027746_row n where
   f 2 = 3; f 3 = 2; f p = p
-- Reinhard Zumkeller, Apr 09 2012, Jan 03 2011

Mathematica

a[n_] := Times @@ Power @@@ (FactorInteger[n] /. {2, e2_} -> {0, e2} /. {3, e3_} -> {2, e3} /. {0, e2_} -> {3, e2}); Table[a[n], {n, 1, 69}] (* Jean-François Alcover, Nov 20 2012 *)
a[n_] := n * Times @@ ({3/2, 2/3}^IntegerExponent[n, {2, 3}]); Array[a, 100] (* Amiram Eldar, Sep 20 2020 *)

PARI

a(n)=my(x=valuation(n, 2)-valuation(n, 3)); n*2^-x*3^x \\ Charles R Greathouse IV, Jun 28 2015

Python

from operator import mul
from functools import reduce
from sympy import factorint
def A064614(n):
    return reduce(mul,((5-p if 2<=p<=3 else p)**e for p,e in factorint(n).items())) if n > 1 else n
# Chai Wah Wu, Dec 27 2014

Formula

a(n) = A065330(n) * (2 ^ A007949(n)) * (3 ^ A007814(n)). - Reinhard Zumkeller, Jan 03 2011

Completely multiplicative with a(2) = 3, a(3) = 2, and a(p) = p for primes p > 3. - Charles R Greathouse IV, Jun 28 2015

Sum_{k=1..n} a(k) ~ (6/7) * n^2. - Amiram Eldar, Oct 28 2022

Dirichlet g.f.: zeta(s-1)*((2^s-2)*(3^s-3))/((2^s-3)*(3^s-2)). - Amiram Eldar, Dec 30 2022