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 A053575 Odd part of phi(n): a(n) = A000265(A000010(n)). 9
 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, 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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS 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). Multiplicative with a(2^e) = 1, a(p^e) = p^(e-1)*A000265(p-1). - Christian G. Bower, May 16 2005 LINKS Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 FORMULA From Bob Selcoe, Aug 22 2017: (Start) 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: a(1) = 1; a(n) = (n-1)" when n is prime; and a(n) = Product_{i = 1..z} a(p_i)*n"/S when n is composite (see Examples). (End) EXAMPLE 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] From Bob Selcoe, Aug 22 2017: (Start) a(89) = 88/8 = 11. 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. (End) MAPLE a:= n-> (t-> t/2^padic[ordp](t, 2))(numtheory[phi](n)): seq(a(n), n=1..80);  # Alois P. Heinz, Apr 14 2020 MATHEMATICA Array[NestWhile[Ceiling[#/2] &, EulerPhi@ #, EvenQ] &, 94] (* Michael De Vlieger, Aug 22 2017 *) (* or *) t=Array[EulerPhi, 94]; t/2^IntegerExponent[t, 2] (* Giovanni Resta, Aug 23 2017 *) PROG (PARI) a(n)=n=eulerphi(n); n>>valuation(n, 2) \\ Charles R Greathouse IV, Mar 05 2013 (Haskell) a053575 = a000265 . a000010  -- Reinhard Zumkeller, Oct 09 2013 CROSSREFS Cf. A000010, A000265. Cf. A227944. Sequence in context: A035648 A322821 A213621 * A293485 A250207 A216319 Adjacent sequences:  A053572 A053573 A053574 * A053576 A053577 A053578 KEYWORD nonn,mult AUTHOR Labos Elemer, Jan 18 2000 STATUS approved

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Last modified July 11 08:53 EDT 2020. Contains 335626 sequences. (Running on oeis4.)