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A072911 Number of "phi-divisors" of n. 9
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 (list; graph; refs; listen; history; text; internal format)
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

J. Sandor, 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

L. Toth, On certain arithmetic functions involving exponential divisors, Annales Univ. Sci. Budapest., Sect. Comp., 27 (2004), 285-294. [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.

MAPLE

A000010 := proc(n) numtheory[phi](n) ; end: A072911 := proc(n) local ifs, a, p; a := 1 ; ifs := ifactors(n)[2] ; for p in ifs do a := a*A000010(op(2, p)) ; od: RETURN(a) ; end: for n from 1 to 150 do printf("%d, ", A072911(n)) ; od: # R. J. Mathar, Sep 25 2008

MATHEMATICA

a[n_] := Times @@ EulerPhi[FactorInteger[n][[All, 2]]];

Array[a, 105] (* Jean-Fran├žois Alcover, Nov 16 2017 *)

PROG

(Haskell)

a072911 = product . map (a000010 . fromIntegral) . a124010_row

-- Reinhard Zumkeller, Mar 13 2012

CROSSREFS

Cf. A061389.

Cf. A124010, A000010, A049599, A049419.

Sequence in context: A061704 A325837 A050361 * A325988 A328856 A053150

Adjacent sequences:  A072908 A072909 A072910 * A072912 A072913 A072914

KEYWORD

nonn,mult

AUTHOR

Yasutoshi Kohmoto, Aug 21 2002

EXTENSIONS

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

STATUS

approved

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Last modified September 28 12:34 EDT 2020. Contains 337393 sequences. (Running on oeis4.)