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
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.
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-François 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
KEYWORD
nonn,easy,nice,mult
AUTHOR
EXTENSIONS
More terms from Jud McCranie
STATUS
approved