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A204455 Squarefree product of all odd primes dividing n, and 1 if n is a power of 2: A099985/2.

%I A204455 #66 Jul 22 2024 06:32:48
%S A204455 1,1,3,1,5,3,7,1,3,5,11,3,13,7,15,1,17,3,19,5,21,11,23,3,5,13,3,7,29,
%T A204455 15,31,1,33,17,35,3,37,19,39,5,41,21,43,11,15,23,47,3,7,5,51,13,53,3,
%U A204455 55,7,57,29,59,15,61,31,21,1,65,33,67,17,69,35,71,3
%N A204455 Squarefree product of all odd primes dividing n, and 1 if n is a power of 2: A099985/2.
%C A204455 There are no odd primes dividing n iff n is a power of 2.
%C A204455 This sequence coincides with the bisection of A007947 (even indices), which is A099985, dividing out the even prime 2 in the squarefree kernel.
%C A204455 a(n) divides A106609(n) for n>=1. - _Alexander R. Povolotsky_, Apr 06 2015
%H A204455 Michael De Vlieger, <a href="/A204455/b204455.txt">Table of n, a(n) for n = 1..10000</a>
%F A204455 a(n) = A099985(n)/2 = A007947(2*n)/2.
%F A204455 a(n) = A000265(A007947(n)) = A007947(A000265(n)). - _Charles R Greathouse IV_, Jan 19 2012
%F A204455 Multiplicative with a(p^e)=p for p <> 2 and a(2^e)=1. - _R. J. Mathar_, Jul 02 2013
%F A204455 a(n) = Sum_{d|n} phi(d)*mu(2d)^2. - _Ridouane Oudra_, Sep 02 2019
%F A204455 From _Richard L. Ollerton_, May 09 2021: (Start)
%F A204455 a(n) = Sum_{k=1..n} mu(2*n/gcd(n,k))^2.
%F A204455 a(n) = Sum_{k=1..n} mu(2*gcd(n,k))^2*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
%F A204455 Sum_{k=1..n} a(k) ~ c * n^2, where c = (2/5) * Product_{p prime} (1 - 1/(p*(p+1))) = (2/5) * A065463 = 0.281776... . - _Amiram Eldar_, Nov 19 2022
%F A204455 a(n) = Sum_{d divides n, d odd} mu(d)^2 * phi(d). - _Peter Bala_, Feb 01 2024
%e A204455 a(5)=5 because 5 is a single odd prime.
%e A204455 a(9)=3 because 9=3*3 has as squarefree part 3.
%e A204455 a(1)=1 because 1 is a power of 2, having no odd primes as a factor.
%p A204455 A204455 := proc(n)
%p A204455     local p;
%p A204455     numtheory[factorset](n) minus {2} ;
%p A204455     mul(p,p=%) ;
%p A204455 end proc:
%p A204455 seq(A204455(n),n=1..40) ; # _R. J. Mathar_, Jan 25 2017
%t A204455 f[n_] := Select[First /@ FactorInteger@ n, PrimeQ@ # && OddQ@ # &]; Times @@@ (f /@ Range@ 120) (* _Michael De Vlieger_, Apr 08 2015 *)
%o A204455 (PARI) a(n) = {my(f = factor(n)); prod(k=1, #f~, if (f[k,1] % 2, f[k,1], 1));} \\ _Michel Marcus_, Apr 07 2015
%o A204455 (PARI) a(n) = factorback(setminus(factorint(n)[, 1]~, [2])) \\ _Jianing Song_, Aug 09 2022
%Y A204455 Cf. A099985, A099984, A007947, A000265, A065463, A106609.
%K A204455 nonn,mult
%O A204455 1,3
%A A204455 _Wolfdieter Lang_, Jan 19 2012