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A047994 Unitary totient (or unitary phi) function uphi(n). 125
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

A divisor d of n is called a unitary divisor if gcd(d, n/d) = 1. Define gcd*(k,n) to be the largest divisor d of k that is also a unitary divisor of n (that is, such that gcd(d, n/d) = 1). The unitary totient function a(n) = number of k with 1 <= k <= n such that gcd*(k,n) = 1. - N. J. A. Sloane, Aug 08 2021

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.

Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]

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.

L. Toth, A survey of gcd-sum functions, J. Int. Seq. 13 (2010) # 10.8.1.

FORMULA

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

a(n) = A000010(n)*A000203(A003557(n))/A003557(n). - Velin Yanev and Charles R Greathouse IV, Aug 23 2017

From Amiram Eldar, May 29 2020: (Start)

a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(d) * n/d.

Sum_{d|n, gcd(d, n/d) = 1} a(d) = n.

a(n) >= phi(n) = A000010(n), with equality if and only if n is squarefree (A005117). (End)

Sum_{k=1..n} a(k) ~ c * Pi^2 * n^2 / 12, where c = A065464 = Product_{primes p} (1 - 2/p^2 + 1/p^3). - Vaclav Kotesovec, Jun 15 2020

EXAMPLE

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

MAPLE

A047994 := proc(n)

    local a, f;

    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 *)

uphi[n_] := If[n==1, 1, Product[{p, e} = pe; p^e-1, {pe, FactorInteger[n]}] ]; Array[uphi, 80] (* Jean-Fran├žois Alcover, Nov 17 2018 *)

PROG

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

(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - 2*X + p*X^2)/(1-X)/(1-p*X))[n], ", ")) \\ Vaclav Kotesovec, Jun 15 2020

(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

(Python)

from math import prod

from sympy import factorint

def A047994(n): return prod(p**e-1 for p, e in factorint(n).items()) # Chai Wah Wu, Sep 24 2021

CROSSREFS

Cf. A000010, A000203, A001221, A003557, A049865, A003271, A028233, A076479.

Sequence in context: A354985 A344005 A345044 * A193024 A340368 A153038

Adjacent sequences:  A047991 A047992 A047993 * A047995 A047996 A047997

KEYWORD

nonn,easy,nice,mult

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Jud McCranie

STATUS

approved

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Last modified October 6 04:09 EDT 2022. Contains 357261 sequences. (Running on oeis4.)