A327838 -id:A327838 - cp's OEIS frontend

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

Yasutoshi Kohmoto, Aug 21 2002

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

József Sándor, On an exponential totient function, Studia Univ. Babes-Bolyai, Math., 41 (1996), 91-94. [Laszlo Toth, Oct 06 2008]

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]

Cf. A061389, A327838.

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

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

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

a[n_] := Times @@ EulerPhi[FactorInteger[n][[All, 2]]];
Array[a, 105] (* Jean-François Alcover, Nov 16 2017 *)

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

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

A361012 Multiplicative with a(p^e) = sigma(e), where sigma = A000203.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 4, 3, 1, 1, 3, 1, 1, 1, 7, 1, 3, 1, 3, 1, 1, 1, 4, 3, 1, 4, 3, 1, 1, 1, 6, 1, 1, 1, 9, 1, 1, 1, 4, 1, 1, 1, 3, 3, 1, 1, 7, 3, 3, 1, 3, 1, 4, 1, 4, 1, 1, 1, 3, 1, 1, 3, 12, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 3, 3, 1, 1, 1, 7, 7, 1, 1, 3, 1, 1

Offset: 1

Vaclav Kotesovec, Feb 28 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A049419, A145353, A072911, A327837, A327838.

g[p_, e_] := DivisorSigma[1, e]; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(sigma, factor(n)[, 2])); \\ Amiram Eldar, Jan 07 2025

from math import prod
from sympy import divisor_sigma, factorint
def A361012(n): return prod(divisor_sigma(e) for e in factorint(n).values()) # Chai Wah Wu, Feb 28 2023

Dirichlet g.f.: Product_{p prime} (1 + Sum_{e>=1} sigma(e) / p^(e*s)).

Sum_{k=1..n} a(k) ~ c * n, where c = Product_{p prime} (1 + Sum_{e>=2} (sigma(e) - sigma(e-1)) / p^e) = 2.96008030202494141048182047811089469392843909592516341... = A361013

A361013 Decimal expansion of a constant related to the asymptotics of A361012.

Original entry on oeis.org

2, 9, 6, 0, 0, 8, 0, 3, 0, 2, 0, 2, 4, 9, 4, 1, 4, 1, 0, 4, 8, 1, 8, 2, 0, 4, 7, 8, 1, 1, 0, 8, 9, 4, 6, 9, 3, 9, 2, 8, 4, 3, 9, 0, 9, 5, 9, 2, 5, 1, 6, 3, 4, 1, 1, 9, 6, 7, 5, 0, 4, 4, 8, 0, 8, 6, 6, 3, 3, 9, 3, 5, 7, 8, 7, 3, 7, 3, 8, 2, 4, 9, 5, 8, 4, 6, 2, 6, 7, 3, 8, 5, 0, 1, 0, 8, 0, 5, 1, 7, 8, 6, 0, 6, 6

Offset: 1

Vaclav Kotesovec, Feb 28 2023

			2.960080302024941410481820478110894693928439095925163411967504480866339...

Cf. A361012, A327837, A327838.

$MaxExtraPrecision = 1000; smax = 500; Do[Clear[f]; f[p_] := 1 + Sum[(DivisorSigma[1, e] - DivisorSigma[1, e-1])/p^e, {e, 2, emax}]; cc = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, smax}], x, smax + 1]]; Print[f[2] * f[3] * f[5] * f[7] * Exp[N[Sum[cc[[n]]*(PrimeZetaP[n] - 1/2^n - 1/3^n - 1/5^n - 1/7^n), {n, 2, smax}], 120]]], {emax, 100, 1000, 100}]

Equals limit_{n->oo} A361012(n) / n.

Equals Product_{p prime} (1 + Sum_{e>=2} (sigma(e) - sigma(e-1)) / p^e), where sigma = A000203.

A358658 Decimal expansion of the asymptotic mean of the e-unitary Euler function (A321167).

Original entry on oeis.org

1, 3, 0, 7, 3, 2, 1, 3, 7, 1, 7, 0, 6, 0, 7, 2, 3, 6, 9, 2, 9, 6, 4, 2, 2, 8, 0, 4, 2, 5, 3, 9, 8, 8, 3, 9, 1, 4, 2, 7, 4, 3, 4, 6, 8, 6, 0, 8, 2, 3, 9, 4, 0, 9, 8, 0, 1, 5, 3, 6, 3, 5, 6, 9, 8, 1, 7, 0, 0, 9, 7, 0, 8, 9, 0, 0, 8, 4, 9, 7, 3, 2, 2, 0, 0, 7, 2, 0, 2, 5, 4, 0, 4, 5, 4, 8, 4, 4, 8, 1, 2, 9, 7, 2, 9

Offset: 1

Amiram Eldar, Nov 25 2022

			1.307321371706072369296422804253988391427434686082394...

Nicuşor Minculete and László Tóth, Exponential unitary divisors, Annales Univ. Sci. Budapest., Sect. Comp. Vol. 35 (2011), pp. 205-216.

Cf. A047994, A321167, A327838.

f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; $MaxExtraPrecision = 500; m = 500; fun[x_] := Log[1 + Sum[x^e*(uphi[e] - uphi[e - 1]), {e, 3, m}]]; c = Rest[CoefficientList[Series[fun[x], {x, 0, m}], x]*Range[0, m]]; RealDigits[Exp[fun[1/2] + NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k)/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Equals lim_{m->oo} (1/m) Sum_{k=1..m} A321167(k).

Equals Product_{p prime} (1 + Sum_{e >= 3} (uphi(e) - uphi(e-1))/p^e), where uphi is the unitary totient function (A047994).

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A072911 Number of "phi-divisors" of n.

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A361012 Multiplicative with a(p^e) = sigma(e), where sigma = A000203.

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A361013 Decimal expansion of a constant related to the asymptotics of A361012.

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A358658 Decimal expansion of the asymptotic mean of the e-unitary Euler function (A321167).

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