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%I A053575 #45 Dec 27 2020 10:09:30
%S A053575 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 A053575 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 A053575 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 A053575 Odd part of phi(n): a(n) = A000265(A000010(n)).
%C A053575 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 A053575 Multiplicative with a(2^e) = 1, a(p^e) = p^(e-1)*A000265(p-1). - _Christian G. Bower_, May 16 2005
%H A053575 Reinhard Zumkeller, <a href="/A053575/b053575.txt">Table of n, a(n) for n = 1..10000</a>
%F A053575 From _Bob Selcoe_, Aug 22 2017: (Start)
%F A053575 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 A053575 a(1) = 1;
%F A053575 a(n) = (n-1)" when n is prime; and
%F A053575 a(n) = Product_{i = 1..z} a(p_i)*n"/S when n is composite (see Examples).
%F A053575 (End)
%F A053575 From _Antti Karttunen_, Dec 27 2020: (Start)
%F A053575 a(n) = A336466(n) for squarefree n (see A005117).
%F A053575 A336466(a(n)) = A336468(n), A329697(a(n)) = A336469(n) = A329697(n) - A005087(n).
%F A053575 (End)
%e A053575 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 A053575 From _Bob Selcoe_, Aug 22 2017: (Start)
%e A053575 a(89) = 88/8 = 11.
%e A053575 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 A053575 (End)
%p A053575 a:= n-> (t-> t/2^padic[ordp](t, 2))(numtheory[phi](n)):
%p A053575 seq(a(n), n=1..80);  # _Alois P. Heinz_, Apr 14 2020
%t A053575 Array[NestWhile[Ceiling[#/2] &, EulerPhi@ #, EvenQ] &, 94] (* _Michael De Vlieger_, Aug 22 2017 *) (* or *)
%t A053575 t=Array[EulerPhi,94]; t/2^IntegerExponent[t,2] (* _Giovanni Resta_, Aug 23 2017 *)
%o A053575 (PARI) a(n)=n=eulerphi(n);n>>valuation(n,2) \\ _Charles R Greathouse IV_, Mar 05 2013
%o A053575 (Haskell)
%o A053575 a053575 = a000265 . a000010  -- _Reinhard Zumkeller_, Oct 09 2013
%Y A053575 Cf. A000010, A000265, A227944, A329697, A336466, A336468, A336469, A339879, A339971, A339974.
%K A053575 nonn,mult
%O A053575 1,7
%A A053575 _Labos Elemer_, Jan 18 2000

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