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A047994 Unitary totient (or unitary phi) function uphi(n). 44
1, 1, 2, 3, 4, 2, 6, 7, 8, 4, 10, 6, 12, 6, 8, 15, 16, 8, 18, 12, 12, 10, 22, 14, 24, 12, 26, 18, 28, 8, 30, 31, 20, 16, 24, 24, 36, 18, 24, 28, 40, 12, 42, 30, 32, 22, 46, 30, 48, 24, 32, 36, 52, 26, 40, 42, 36, 28, 58, 24, 60, 30, 48, 63, 48, 20, 66, 48, 44, 24, 70 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Unitary convolution of A076479 and A000027. - R. J. Mathar, Apr 13 2011

Multiplicative with a(p^e) = p^e - 1. - N. J. A. Sloane, Apr 30 2013

LINKS

T. D. Noe, Table of n, a(n) for n=1..10000

Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Zeitschr. 74 (1960) 66-80

S. R. Finch, Unitarism and infinitarism.

M. Lal, Iterates of the unitary totient function, Math. Comp., 28 (1974), 301-302.

R. J. Mathar, Survey of Dirichlet Series of Multiplicative Arithmetic Functions, arXiv:1106.4038 [math.NT], 2011, Remark 43.

L. Toth, On the Bi-Unitary Analogues of Euler's Arithmetical Function and the Gcd-Sum Function, JIS 12 (2009) 09.5.2

FORMULA

If n = Product p_i^e_i, uphi(n) = Product (p_i^e_i - 1).

EXAMPLE

a(12) = a(3)*a(4) = 2*3 = 6.

MAPLE

A047994 := proc(n)

    local a;

    a := 1 ;

    for f in ifactors(n)[2] do

        a := a*(op(1, f)^op(2, f)-1) ;

    end do:

    a ;

end proc:

seq(A047994(n), n=1..20) ; # R. J. Mathar, Dec 22 2011

MATHEMATICA

uphi[n_] := (Times @@ (Table[ #[[1]]^ #[[2]] - 1, {1} ] & /@ FactorInteger[n]))[[1]]; Table[ uphi[n], {n, 2, 75}] (* Robert G. Wilson v, Sep 06 2004 *)

PROG

(PARI) A047994(n)=prod(i=1, #n=factor(n)~, n[1, i]^n[2, i]-1)

(Haskell)

a047994 n = f n 1 where

   f 1 uph = uph

   f x uph = f (x `div` sppf) (uph * (sppf - 1)) where sppf = a028233 x

-- Reinhard Zumkeller, Aug 17 2011

CROSSREFS

Cf. A049865, A003271, A028233.

Sequence in context: A278963 A178970 A172054 * A193024 A153038 A220335

Adjacent sequences:  A047991 A047992 A047993 * A047995 A047996 A047997

KEYWORD

nonn,easy,nice,mult,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Jud McCranie

STATUS

approved

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Last modified May 27 22:57 EDT 2017. Contains 287210 sequences.