A361013 -id:A361013 - cp's OEIS frontend

A327837 Decimal expansion of the asymptotic mean of the number of exponential divisors function (A049419).

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

Offset: 1

Amiram Eldar, Sep 27 2019

			1.602317102305418052349626315621161003776939495785572...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 52 (constant Z3).
V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions - II, Indian J. Pure Appl. Math., Vol. 11 (1980), pp. 1334-1355 (eq. 2.37 and 3.18, pp. 1346 and 1354).
Abdelhakim Smati and Jie Wu, On the exponential divisor function, Publications de l'Institut Mathématique, Vol. 61 (1997), pp. 21-32.
László Tóth, An order result for the exponential divisor function, Publ. Math. Debrecen, Vol. 71, No. 1-2 (2007), pp. 165-171, arXiv preprint,, arXiv:0708.3552 [math.NT], 2007.
László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1 (section 4.10, p. 30).
Jie Wu, Problème de diviseurs exponentiels et entiers exponentiellement sans facteur carré, Journal de théorie des nombres de Bordeaux, Vol. 7, No. 1, (1995), pp. 133-141.

Cf. A000005, A047994, A049419, A145353, A361013, A361967, A379517, A379518.

Cf. A059956 (constant for unitary divisors), A306071 (bi-unitary), A327576 (infinitary).

$MaxExtraPrecision = 1500; m = 1500; em = 500; f[x_] := 1 + Log[1 + Sum[x^e * (DivisorSigma[0, e] - DivisorSigma[0, e - 1]), {e, 2, em}]]; c = Rest[ CoefficientList[Series[f[x], {x, 0, m}], x] * Range[0, m] ]; RealDigits[ Exp[NSum[Indexed[c, k] * PrimeZetaP[k]/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Equals lim_{k->oo} A145353(k)/k.
Equals Product_{p prime} (1 + Sum_{e >= 2} p^(-e) * (d(e) - d(e-1))), where d(e) is the number of divisors of e (A000005).
Equals Product_{p prime} (1 - 1/p) * (2 - (log(p-1) + QPolyGamma(0, 1, 1/p)) / log(p)). - Vaclav Kotesovec, Feb 27 2023
From Amiram Eldar, Dec 24 2024: (Start)
Equals lim_{m->oo} (1/m) * Sum_{k=1..m} k/uphi(k) = lim_{m->oo} (1/m) * Sum_{k=1..m} A319677(k)/A319676(k), where uphi(k) is the unitary totient function (A047994).
Equals lim_{m->oo} (1/log(m)) * Sum_{k=1..m} 1/uphi(k) = lim_{m->oo} (1/log(m)) * A379517(m)/A379518(m).
Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A361967(k).
Equals Product_{p prime} ((1-1/p) * (1 + Sum_{k>=1} 1/(p^k-1))).
Equals Product_{p prime} (1 + (1-1/p) * Sum_{k>=1} 1/(p^k*(p^k-1))). (End)

More digits from Vaclav Kotesovec, Jun 13 2021

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

A327838 Decimal expansion of the asymptotic mean of the exponential totient function (A072911).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 27 2019

			1.252707785375446126053750751934283060439237967108915...

László Tóth, On certain arithmetic functions involving exponential divisors, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 24 (2004), pp. 285-294.

Cf. A000010, A072911, A322887, A327837, A361013.

$MaxExtraPrecision = 500; m = 500; f[x_] := Log[1 + Sum[x^e * (EulerPhi[e] - EulerPhi[e - 1]), {e, 3, m}]]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x]*Range[0, m]]; RealDigits[Exp[f[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} A072911(k).

Equals Product_{p prime} (1 + Sum_{e >= 3} (phi(e) - phi(e-1))/p^e), where phi is the Euler totient function (A000010).

A361063 Multiplicative with a(p^e) = sigma_2(e), where sigma_2 = A001157.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 10, 5, 1, 1, 5, 1, 1, 1, 21, 1, 5, 1, 5, 1, 1, 1, 10, 5, 1, 10, 5, 1, 1, 1, 26, 1, 1, 1, 25, 1, 1, 1, 10, 1, 1, 1, 5, 5, 1, 1, 21, 5, 5, 1, 5, 1, 10, 1, 10, 1, 1, 1, 5, 1, 1, 5, 50, 1, 1, 1, 5, 1, 1, 1, 50, 1, 1, 5, 5, 1, 1, 1, 21, 21, 1, 1, 5

Offset: 1

Vaclav Kotesovec, Mar 01 2023

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

Cf. A001157, A049419, A361012, A361064.

Cf. A327837, A361013.

g[p_, e_] := DivisorSigma[2, e]; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

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

from math import prod
from sympy import factorint, divisor_sigma
def A361063(n): return prod(divisor_sigma(e,2) for e in factorint(n).values()) # Chai Wah Wu, Mar 01 2023

Dirichlet g.f.: Product_{primes p} (1 + Sum_{e>=1} sigma_2(e) / p^(e*s)).

Sum_{k=1..n} a(k) ~ c * n, where c = Product_{p prime} (1 + Sum_{e>=2} (sigma_2(e) - sigma_2(e-1)) / p^e) = 11.343154585178523783556367128387762286267199879648613456124659589127638983...

A361064 Multiplicative with a(p^e) = sigma_3(e), where sigma_3 = A001158.

Original entry on oeis.org

1, 1, 1, 9, 1, 1, 1, 28, 9, 1, 1, 9, 1, 1, 1, 73, 1, 9, 1, 9, 1, 1, 1, 28, 9, 1, 28, 9, 1, 1, 1, 126, 1, 1, 1, 81, 1, 1, 1, 28, 1, 1, 1, 9, 9, 1, 1, 73, 9, 9, 1, 9, 1, 28, 1, 28, 1, 1, 1, 9, 1, 1, 9, 252, 1, 1, 1, 9, 1, 1, 1, 252, 1, 1, 9, 9, 1, 1, 1, 73, 73, 1, 1, 9

Offset: 1

Vaclav Kotesovec, Mar 01 2023

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

Cf. A001158, A049419, A361012, A361063.

Cf. A327837, A361013.

g[p_, e_] := DivisorSigma[3, e]; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> sigma(x, 3), factor(n)[, 2])); \\ Amiram Eldar, Jan 07 2025

from math import prod
from sympy import factorint, divisor_sigma
def A361064(n): return prod(divisor_sigma(e,3) for e in factorint(n).values()) # Chai Wah Wu, Mar 01 2023

Dirichlet g.f.: Product_{primes p} (1 + Sum_{e>=1} sigma_3(e) / p^(e*s)).

Sum_{k=1..n} a(k) ~ c * n, where c = Product_{p prime} (1 + Sum_{e>=2} (sigma_3(e) - sigma_3(e-1)) / p^e) = 136.775196585091127831467103699999450735835551529525277016916082455332230986...

cp's OEIS Frontend

A327837 Decimal expansion of the asymptotic mean of the number of exponential divisors function (A049419).

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

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A327838 Decimal expansion of the asymptotic mean of the exponential totient function (A072911).

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A361063 Multiplicative with a(p^e) = sigma_2(e), where sigma_2 = A001157.

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A361064 Multiplicative with a(p^e) = sigma_3(e), where sigma_3 = A001158.

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