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A053575 Odd part of phi(n): a(n) = A000265(A000010(n)). 5
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)

Ex 1: a(89) = 88/8 = 11.

Ex 2: 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)

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: A289617 A035648 A213621 * A250207 A216319 A218355

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 September 19 21:54 EDT 2017. Contains 292250 sequences.