A053575 Odd part of phi(n): a(n) = A000265(A000010(n)).

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

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)

From Antti Karttunen, Dec 27 2020: (Start)

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

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

(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, A227944, A329697, A336466, A336468, A336469, A339879, A339971, A339974.

Sequence in context: A369069 A322821 A213621 * A293485 A250207 A216319

Adjacent sequences: A053572 A053573 A053574 * A053576 A053577 A053578

Keyword

nonn,mult

Author

Labos Elemer, Jan 18 2000

Status

approved