A053575 - OEIS

%I #45 Dec 27 2020 10:09:30

%S 1,1,1,1,1,1,3,1,3,1,5,1,3,3,1,1,1,3,9,1,3,5,11,1,5,3,9,3,7,1,15,1,5,

%T 1,3,3,9,9,3,1,5,3,21,5,3,11,23,1,21,5,1,3,13,9,5,3,9,7,29,1,15,15,9,

%U 1,3,5,33,1,11,3,35,3,9,9,5,9,15,3,39,1,27,5,41,3,1,21,7,5,11,3,9,11,15,23

%N Odd part of phi(n): a(n) = A000265(A000010(n)).

%C This is not necessarily the squarefree kernel. E.g., for n=19, phi(19)=18 is divisible by 9, an odd square. Values at which this kernel is 1 correspond to A003401 (polygons constructible with ruler and compass).

%C Multiplicative with a(2^e) = 1, a(p^e) = p^(e-1)*A000265(p-1). - _Christian G. Bower_, May 16 2005

%H Reinhard Zumkeller, <a href="/A053575/b053575.txt">Table of n, a(n) for n = 1..10000</a>

%F From _Bob Selcoe_, Aug 22 2017: (Start)

%F Let n" be the odd part of n, S be the odd squarefree kernel of n and p_i {i = 1..z} be all the prime factors of S. Then the sequence can be constructed by the following:

%F a(1) = 1;

%F a(n) = (n-1)" when n is prime; and

%F a(n) = Product_{i = 1..z} a(p_i)*n"/S when n is composite (see Examples).

%F (End)

%F From _Antti Karttunen_, Dec 27 2020: (Start)

%F a(n) = A336466(n) for squarefree n (see A005117).

%F A336466(a(n)) = A336468(n), A329697(a(n)) = A336469(n) = A329697(n) - A005087(n).

%F (End)

%e n = 70 = 2*5*7, phi(70) = 24 = 8*3, so the odd kernel of phi(70) is a(70)=3. [corrected by _Bob Selcoe_, Aug 22 2017]

%e From _Bob Selcoe_, Aug 22 2017: (Start)

%e a(89) = 88/8 = 11.

%e For n = 8820, 8820 = 2^2*3^2*5*7^2; S = 3*5*7 = 105, n" = 3^2*5*7^2 = 2205. a(3)*a(5)*a(7) = 1*1*3 = 3; a(8820) = 3*2205/105 = 63.

%e (End)

%p a:= n-> (t-> t/2^padic[ordp](t, 2))(numtheory[phi](n)):

%p seq(a(n), n=1..80);  # _Alois P. Heinz_, Apr 14 2020

%t Array[NestWhile[Ceiling[#/2] &, EulerPhi@ #, EvenQ] &, 94] (* _Michael De Vlieger_, Aug 22 2017 *) (* or *)

%t t=Array[EulerPhi,94]; t/2^IntegerExponent[t,2] (* _Giovanni Resta_, Aug 23 2017 *)

%o (PARI) a(n)=n=eulerphi(n);n>>valuation(n,2) \\ _Charles R Greathouse IV_, Mar 05 2013

%o (Haskell)

%o a053575 = a000265 . a000010  -- _Reinhard Zumkeller_, Oct 09 2013

%Y Cf. A000010, A000265, A227944, A329697, A336466, A336468, A336469, A339879, A339971, A339974.

%K nonn,mult

%O 1,7

%A _Labos Elemer_, Jan 18 2000