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A047994
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Unitary totient (or unitary phi) function uphi(n).
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142
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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
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OFFSET
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1,3
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COMMENTS
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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
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LINKS
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FORMULA
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If n = Product p_i^e_i, uphi(n) = Product (p_i^e_i - 1).
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
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EXAMPLE
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a(12) = a(3)*a(4) = 2*3 = 6.
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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
(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
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CROSSREFS
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KEYWORD
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nonn,easy,nice,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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