A072911 Number of "phi-divisors" of n.

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, 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, 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

Offset: 1

Author

Yasutoshi Kohmoto, Aug 21 2002

Keywords

• nonn
• mult

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]

Crossrefs

Cf. A061389, A327838.

Cf. A124010, A000010, A049599, A049419, A361012.

Programs

Haskell

a072911 = product . map (a000010 . fromIntegral) . a124010_row
-- Reinhard Zumkeller, Mar 13 2012

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 *)

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

Extensions

More terms from R. J. Mathar, Sep 25 2008