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

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice
• mult

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

Examples

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

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
• Steven 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.
• László Tóth, On the Bi-Unitary Analogues of Euler's Arithmetical Function and the Gcd-Sum Function, JIS 12 (2009), Article 09.5.2.
• László Tóth, A survey of gcd-sum functions, J. Int. Seq. 13 (2010), Article 10.8.1.

Crossrefs

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

Programs

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

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-François Alcover, Nov 17 2018 *)

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

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

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

Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 2/p^s + 1/p^(2*s-1)). - Amiram Eldar, May 22 2025

Extensions

More terms from Jud McCranie