cp's OEIS Frontend

A072911 Number of "phi-divisors" of n.

%I A072911 #44 Mar 13 2024 07:52:59
%S A072911 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,4,1,1,
%T A072911 1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,1,1,1,1,2,1,1,1,1,
%U A072911 1,1,1,2,1,1,1,1,1,1,1,2,2,1,1,1,1,1,1,2,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,2,1
%N A072911 Number of "phi-divisors" of n.
%C A072911 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.
%C A072911 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
%D A072911 József Sándor, On an exponential totient function, Studia Univ. Babes-Bolyai, Math., 41 (1996), 91-94. [_Laszlo Toth_, Oct 06 2008]
%H A072911 Reinhard Zumkeller, <a href="/A072911/b072911.txt">Table of n, a(n) for n = 1..10000</a>
%H A072911 László Tóth, <a href="https://arxiv.org/abs/math/0610274">On certain arithmetic functions involving exponential divisors</a>, Annales Univ. Sci. Budapest., Sect. Comp., 27 (2004), 285-294; arXiv:math/0610274 [math.NT], 2006-2009. [From _Laszlo Toth_, Oct 06 2008]
%F A072911 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.
%F A072911 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
%p A072911 A072911 := proc(n)
%p A072911     local a, p;
%p A072911     a := 1 ;
%p A072911     for p in ifactors(n)[2] do
%p A072911         a := a*numtheory[phi](op(2, p)) ;
%p A072911     od:
%p A072911     a ;
%p A072911 end:
%p A072911 seq(A072911(n),n=1..100) ; # _R. J. Mathar_, Sep 25 2008
%t A072911 a[n_] := Times @@ EulerPhi[FactorInteger[n][[All, 2]]];
%t A072911 Array[a, 105] (* _Jean-François Alcover_, Nov 16 2017 *)
%o A072911 (Haskell)
%o A072911 a072911 = product . map (a000010 . fromIntegral) . a124010_row
%o A072911 -- _Reinhard Zumkeller_, Mar 13 2012
%Y A072911 Cf. A061389, A327838.
%Y A072911 Cf. A124010, A000010, A049599, A049419, A361012.
%K A072911 nonn,mult
%O A072911 1,8
%A A072911 _Yasutoshi Kohmoto_, Aug 21 2002
%E A072911 More terms from _R. J. Mathar_, Sep 25 2008