OFFSET
1,8
COMMENTS
If n = Product p(i)^r(i), d = Product p(i)^s(i), = s(i)<=r(i) and gcd(s(i), r(i)) = 1, then d is a phi-divisor of n.
The integers n = Product_{i=1..r} p_i^{a_i} and m = Product_{i=1..r} p_i^{b_i}, a_i, b_i >= 1 (1 <= i <= r) having the same prime factors are called exponentially coprime, if gcd(a_i, b_i) = 1 for every 1 <= i <= r, i.e., the only common exponential divisor of n and m is Product_{i=1..r} p_i = the common squarefree kernel of n and m, cf. A049419, A007947. The terms of this sequence count the divisors d of n such that d and n are exponentially coprime. - Laszlo Toth, Oct 06 2008
REFERENCES
József Sándor, On an exponential totient function, Studia Univ. Babes-Bolyai, Math., 41 (1996), 91-94. [Laszlo Toth, Oct 06 2008]
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
László Tóth, On certain arithmetic functions involving exponential divisors, Annales Univ. Sci. Budapest., Sect. Comp., 27 (2004), 285-294; arXiv:math/0610274 [math.NT], 2006-2009. [From Laszlo Toth, Oct 06 2008]
FORMULA
If n = Product p(i)^r(i) then a(n) = Product (phi(r(i))), where phi(k) is the Euler totient function of k, cf. A000010.
Sum_{k=1..n} a(k) ~ c_1 * n + c_2 * n^(1/3) + O(n^(1/5+eps)), where c_1 = A327838 (Tóth, 2004). - Amiram Eldar, Oct 30 2022
MAPLE
A072911 := proc(n)
local a, p;
a := 1 ;
for p in ifactors(n)[2] do
a := a*numtheory[phi](op(2, p)) ;
od:
a ;
end:
seq(A072911(n), n=1..100) ; # R. J. Mathar, Sep 25 2008
MATHEMATICA
a[n_] := Times @@ EulerPhi[FactorInteger[n][[All, 2]]];
Array[a, 105] (* Jean-François Alcover, Nov 16 2017 *)
PROG
(Haskell)
a072911 = product . map (a000010 . fromIntegral) . a124010_row
-- Reinhard Zumkeller, Mar 13 2012
KEYWORD
nonn,mult
AUTHOR
Yasutoshi Kohmoto, Aug 21 2002
EXTENSIONS
More terms from R. J. Mathar, Sep 25 2008
STATUS
approved