login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A047994 Unitary totient (or unitary phi) function uphi(n). 142
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
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, 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-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
Sequence in context: A354985 A344005 A345044 * A193024 A340368 A367175
KEYWORD
nonn,easy,nice,mult
AUTHOR
EXTENSIONS
More terms from Jud McCranie
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)