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A064614 Exchange 2 and 3 in the prime factorization of n. 14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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
LINKS
A. B. Frizell, Certain non-enumerable sets of infinite permutations, Bull. Amer. Math. Soc. 21, No. 10 (1915), 495-499.
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
EXAMPLE
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.
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 *)
PROG
(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
(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
(PARI) a(n)=my(x=valuation(n, 2)-valuation(n, 3)); n*2^-x*3^x \\ Charles R Greathouse IV, Jun 28 2015
CROSSREFS
Sequence in context: A268824 A306470 A251560 * A234747 A016650 A033313
KEYWORD
nonn,mult,nice,easy
AUTHOR
Reinhard Zumkeller, Sep 25 2001
STATUS
approved

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Last modified February 22 15:38 EST 2024. Contains 370256 sequences. (Running on oeis4.)