A181351 - OEIS

%I #24 Oct 14 2024 23:57:30

%S 1,5,3,25,2,15,7,125,9,10,11,75,13,35,6,625,17,45,19,50,21,55,23,375,

%T 4,65,27,175,29,30,31,3125,33,85,14,225,37,95,39,250,41,105,43,275,18,

%U 115,47,1875,49,20,51,325,53,135,22,875,57,145,59,150,61,155,63

%N Exchange 2 and 5 in the prime factorization of n.

%C A self-inverse permutation of the natural numbers.

%C a(1) = 1, a(2) = 5, a(5) = 2, a(p) = p for primes p = 3 and p > 5 and a(u * v) = a(u) * a(v) for u, v > 0.

%C A permutation of the natural numbers: a(a(n)) = n for all n and a(n) = n if and only if n = 10^k * m for k >= 0 and m > 0 with GCD(m, 10) = 1. This is to (2,5) as A064614 is to (2,3).

%H Amiram Eldar, <a href="/A181351/b181351.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.

%F Dirichlet g.f.: zeta(s-1)*(2^s-2)*(5^s-5)/((2^s-5)*(5^s-2)). - _Amiram Eldar_, Jul 18 2023

%e a(15) = a(3*5) = a(3)*a(5) = 3*2 = 6.

%e a(16) = a(2^4) = a(2)^4 = 5^4 = 625.

%t a[n_] := n * Times @@ ({5/2, 2/5}^IntegerExponent[n, {2, 5}]); Array[a, 100] (* _Amiram Eldar_, Jul 18 2023 *)

%o (PARI) a(n)=n*(5/2)^valuation(n,2)*(2/5)^valuation(n,5) \\ _Charles R Greathouse IV_, Dec 07 2011

%Y Cf. A064614.

%K nonn,easy,mult

%O 1,2

%A _Jonathan Vos Post_, Jan 29 2011

%E a(20) corrected by _Charles R Greathouse IV_, Dec 07 2011