cp's OEIS Frontend

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

%I A064614 #52 Dec 30 2022 03:55:05
%S A064614 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,
%T A064614 39,8,63,29,30,31,243,22,51,35,36,37,57,26,135,41,42,43,99,20,69,47,
%U A064614 162,49,75,34,117,53,24,55,189,38,87,59,90,61,93,28,729,65,66,67,153,46
%N A064614 Exchange 2 and 3 in the prime factorization of n.
%C A064614 A self-inverse permutation of the natural numbers.
%C A064614 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.
%C A064614 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).
%C A064614 A000244 and A000079 give record values and where they occur. - _Reinhard Zumkeller_, Feb 08 2010
%H A064614 T. D. Noe, <a href="/A064614/b064614.txt">Table of n, a(n) for n = 1..1000</a>
%H A064614 A. B. Frizell, <a href="http://dx.doi.org/10.1090/S0002-9904-1915-02686-8">Certain non-enumerable sets of infinite permutations</a>, Bull. Amer. Math. Soc. 21, No. 10 (1915), 495-499.
%H A064614 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%H A064614 <a href="/index/Di#divseq">Index to divisibility sequences</a>.
%F A064614 a(n) = A065330(n) * (2 ^ A007949(n)) * (3 ^ A007814(n)). - _Reinhard Zumkeller_, Jan 03 2011
%F A064614 Completely multiplicative with a(2) = 3, a(3) = 2, and a(p) = p for primes p > 3. - _Charles R Greathouse IV_, Jun 28 2015
%F A064614 Sum_{k=1..n} a(k) ~ (6/7) * n^2. - _Amiram Eldar_, Oct 28 2022
%F A064614 Dirichlet g.f.: zeta(s-1)*((2^s-2)*(3^s-3))/((2^s-3)*(3^s-2)). - _Amiram Eldar_, Dec 30 2022
%e A064614 a(15) = a(3*5) = a(3)*a(5) = 2*5 = 10;
%e A064614 a(16) = a(2^4) = a(2)^4 = 3^4 = 81;
%e A064614 a(17) = 17;
%e A064614 a(18) = a(2*3^2) = a(2)*a(3^2) = 3*a(3)^2 = 3*2^2 = 12.
%t A064614 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 *)
%t A064614 a[n_] := n * Times @@ ({3/2, 2/3}^IntegerExponent[n, {2, 3}]); Array[a, 100] (* _Amiram Eldar_, Sep 20 2020 *)
%o A064614 (Haskell)
%o A064614 a064614 1 = 1
%o A064614 a064614 n = product $ map f $ a027746_row n where
%o A064614    f 2 = 3; f 3 = 2; f p = p
%o A064614 -- _Reinhard Zumkeller_, Apr 09 2012, Jan 03 2011
%o A064614 (Python)
%o A064614 from operator import mul
%o A064614 from functools import reduce
%o A064614 from sympy import factorint
%o A064614 def A064614(n):
%o A064614     return reduce(mul,((5-p if 2<=p<=3 else p)**e for p,e in factorint(n).items())) if n > 1 else n
%o A064614 # _Chai Wah Wu_, Dec 27 2014
%o A064614 (PARI) a(n)=my(x=valuation(n, 2)-valuation(n, 3)); n*2^-x*3^x \\ _Charles R Greathouse IV_, Jun 28 2015
%Y A064614 Cf. A000079, A000244, A007814, A007949, A027746, A064615, A065330, A253046, A253047.
%K A064614 nonn,mult,nice,easy
%O A064614 1,2
%A A064614 _Reinhard Zumkeller_, Sep 25 2001