A327837 -id:A327837 - cp's OEIS frontend

A049419 a(1) = 1; for n > 1, a(n) = number of exponential divisors of n.

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 4, 1, 1

Offset: 1

Yasutoshi Kohmoto

The exponential divisors of a number x = Product p(i)^r(i) are all numbers of the form Product p(i)^s(i) where s(i) divides r(i) for all i.
Wu gives a complicated Dirichlet g.f.
a(1) = 1 by convention. This is also required for a function to be multiplicative. - N. J. A. Sloane, Mar 03 2009
The inverse Moebius transform seems to be in A124315. The Dirichlet inverse appears to be related to A166234. - R. J. Mathar, Jul 14 2014

			a(8)=2 because 2 and 2^3 are e-divisors of 8.
The sets of e-divisors start as:
  1:{1}
  2:{2}
  3:{3}
  4:{2, 4}
  5:{5}
  6:{6}
  7:{7}
  8:{2, 8}
  9:{3, 9}
  10:{10}
  11:{11}
  12:{6, 12}
  13:{13}
  14:{14}
  15:{15}
  16:{2, 4, 16}
  17:{17}
  18:{6, 18}
  19:{19}
  20:{10, 20}
  21:{21}
  22:{22}
  23:{23}
  24:{6, 24}

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Andrew V. Lelechenko, Exponential and infinitary divisors, arXiv:1405.7597 [math.NT], 2014, sequence tau^(e).
David Moews, A database of aliquot cycles.
J. O. M. Pedersen, Tables of Aliquot Cycles.
J. O. M. Pedersen, Tables of Aliquot Cycles. [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Cached copy, pdf file only]
László Tóth and Nicuşor Minculete, Exponential unitary divisors, arXiv:0910.2798 [math.NT], 2009.
Tim Trudgian, The sum of the unitary divisor function, arXiv:1312.4615 [math.NT], 2013-2014, Section 3.
Eric Weisstein's World of Mathematics, e-Divisor.
Jie Wu, Problème de diviseurs exponentiels et entiers exponentiellement sans facteur carré, J. Theor. Nombr. Bordeaux 7 (1) (1995) 133-141.

Row lengths of A322791.
Cf. A049599, A061389, A051377 (sum of e-divisors).
Partial sums are in A099593.
Cf. A124010, A000005, A049599, A072911, A124315, A166234, A327837, A361012.

A049419:=n->Product(List(Collected(Factors(n)), p -> Tau(p[2]))); List([1..10^4], n -> A049419(n)); # Muniru A Asiru, Oct 29 2017

a049419 = product . map (a000005 . fromIntegral) . a124010_row
-- Reinhard Zumkeller, Mar 13 2012

A049419 := proc(n)
    local a;
    a := 1 ;
    for pf in ifactors(n)[2] do
        a := a*numtheory[tau](op(2,pf)) ;
    end do:
    a ;
end proc:
seq(A049419(n),n=1..20) ; # R. J. Mathar, Jul 14 2014

a[1] = 1; a[p_?PrimeQ] = 1; a[p_?PrimeQ, e_] := DivisorSigma[0, e]; a[n_] := Times @@ (a[#[[1]], #[[2]]] & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, Jan 30 2012, after Vladeta Jovovic *)

a(n) = vecprod(apply(numdiv, factor(n)[,2])); \\ Amiram Eldar, Mar 27 2023

Multiplicative with a(p^e) = tau(e). - Vladeta Jovovic, Jul 23 2001

Sum_{k=1..n} a(k) ~ A327837 * n. - Vaclav Kotesovec, Feb 27 2023

More terms from Jud McCranie, May 29 2000

A361967 Number of numbers k such that uphi(k) = n, where uphi is the unitary totient function (A047994).

Original entry on oeis.org

2, 2, 1, 2, 0, 3, 1, 4, 0, 2, 0, 5, 0, 1, 1, 2, 0, 3, 0, 2, 0, 2, 0, 8, 0, 2, 0, 3, 0, 4, 1, 4, 0, 0, 0, 6, 0, 0, 0, 4, 0, 3, 0, 2, 0, 2, 0, 11, 0, 0, 0, 2, 0, 1, 0, 4, 0, 2, 0, 8, 0, 1, 1, 2, 0, 3, 0, 0, 0, 3, 0, 11, 0, 0, 0, 0, 0, 3, 0, 8, 0, 2, 0, 5, 0, 0, 0

Offset: 1

Amiram Eldar, Apr 01 2023

nonn

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

Row lengths of A361966.
The unitary version of A014197.
Cf. A047994, A135347, A327837, A347771 (positions of 0's), A361966, A361968 (indices of records), A361969 (positions of 1's), A361970, A361971 (record values).

a[n_] := Length[invUPhi[n]]; Array[a, 100] (* using the function invUPhi from A361966 *)

a(A347771(n)) = 0.
a(A361969(n)) = 1.
a(A361970(n)) = n.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A327837. - Amiram Eldar, Dec 24 2024

A145353 Sum of the number of e-divisors of all numbers from 1 up to n.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 12, 13, 14, 16, 17, 18, 19, 22, 23, 25, 26, 28, 29, 30, 31, 33, 35, 36, 38, 40, 41, 42, 43, 45, 46, 47, 48, 52, 53, 54, 55, 57, 58, 59, 60, 62, 64, 65, 66, 69, 71, 73, 74, 76, 77, 79, 80, 82, 83, 84, 85, 87, 88, 89, 91, 95, 96, 97, 98, 100, 101, 102, 103, 107

Offset: 1

Jaroslav Krizek and N. J. A. Sloane, Mar 03 2009

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. Wu, Problème de diviseurs exponentiels et entiers exponentiellement sans facteur carré, J. Theor. Nombr. Bordeaux 7 (1) (1995) 133-141.

Equals partial sums of A049419.
Cf. A099353, A327837.
Different from A013936 (which does not contain 52).

f[p_, e_]  := DivisorSigma[0, e]; ediv[n_] := Times @@ (f @@@ FactorInteger[n]); Accumulate[Array[ediv, 100]] (* Amiram Eldar, Jun 23 2019 *)

d(n) = {my(f = factor(n)); prod(i = 1, #f~, numdiv(f[i,2]));}
lista(nmax) = {my(s = 0); for(n = 1, nmax, s += d(n); print1(s, ", ")); } \\ Amiram Eldar, Dec 08 2022

a(n) ~ c * n, where c = A327837. - Amiram Eldar, Dec 08 2022

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.

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).

A379517 Numerators of the partial sums of the reciprocals of the unitary totient function (A047994).

Original entry on oeis.org

1, 2, 5, 17, 37, 43, 15, 109, 225, 239, 1223, 3809, 1293, 4019, 1031, 209, 1693, 1735, 5261, 5345, 5429, 27649, 306659, 310619, 312929, 317549, 4155857, 4195897, 603091, 615961, 619393, 19304143, 19463731, 1228951, 9898103, 4982299, 1251116, 2524397, 10164083

Offset: 1

Amiram Eldar, Dec 24 2024

			Fractions begin with 1, 2, 5/2, 17/6, 37/12, 43/12, 15/4, 109/28, 225/56, 239/56, 1223/280, 3809/840, ...

Amiram Eldar, Table of n, a(n) for n = 1..1000
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018. See p. 52.
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.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.10, pp. 30-31.
Rimer Zurita, Generalized Alternating Sums of Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 23 (2020), Article 20.10.4. See section 4.5, pp. 16-17.

Cf. A001620, A047994, A177754, A370899, A327837, A379518 (denominators), A379519.

uphi[n_] := Times @@ (-1 + Power @@@ FactorInteger[n]); uphi[1] = 1; Numerator[Accumulate[Table[1/uphi[n], {n, 1, 50}]]]

uphi(n) = {my(f = factor(n)); prod(i = 1, #f~, -1 + f[i, 1]^f[i, 2]);}
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / uphi(k); print1(numerator(s), ", "))};

a(n) = numerator(Sum_{k=1..n} 1/A047994(k)).

a(n)/A379518(n) = L * log(n) + M + O(log(n)^(5/3)/n), where L = A327837, M = L * (gamma - B + A1 + A2), gamma = A001620, B = Sum_{p prime} (1-1/p) * log(p) * Sum_{k>=1} k/(p^k*(p^k-1)) / A(p), A1 = Sum_{p prime} log(p)/(p^2*(p-1)*A(p)), A2 = Sum_{p prime} ((A*(p)(p)*log(p)/p^2), A(p) = 1 + (1-1/p) * Sum_{k>=1} 1/(p^k*(p^k-1)), and A*(p) = Sum_{k>=1} 1/(p^k*p^(k+1)-1)*A(p)) (Sita Ramaiah and Suryanarayana, 1980).

A160097 Number of non-exponential divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 2, 1, 3, 1, 4, 1, 3, 3, 2, 1, 4, 1, 4, 3, 3, 1, 6, 1, 3, 2, 4, 1, 7, 1, 4, 3, 3, 3, 5, 1, 3, 3, 6, 1, 7, 1, 4, 4, 3, 1, 7, 1, 4, 3, 4, 1, 6, 3, 6, 3, 3, 1, 10, 1, 3, 4, 3, 3, 7, 1, 4, 3, 7, 1, 8, 1, 3, 4, 4, 3, 7, 1, 7, 2, 3, 1, 10, 3, 3, 3, 6, 1, 10, 3, 4, 3, 3, 3, 10, 1, 4, 4, 5

Offset: 1

Jaroslav Krizek, May 01 2009

nonn

The non-exponential divisors d|n of a number n = Product_i p(i)^e(i) are divisors d not of the form Product_i p(i)^s(i), s(i)|e(i) for all i.

			a(8) = 2 because 1 and 2^2 are non-exponential divisors of 8 = 2^3. 2^2 is a non-exponential divisor because 2^2 = 4 divides 8, but the exponent 2 = s(1) does not divide the exponent 3 = e(1).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A049419, A000040, A006881, A120944, A000961, A053810.

Cf. A001620, A327837.

f1[p_, e_] := e + 1; f2[p_, e_] := DivisorSigma[0, e]; a[1] = 1; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; Array[a, 100] (* Amiram Eldar, Oct 26 2021 *)

A049419(n) = { my(f = factor(n), m = 1); for(k=1, #f~, m *= numdiv(f[k, 2])); m; } \\ After Jovovic's formula for A049419.
A160097(n) = if(1==n,n,(numdiv(n) - A049419(n))); \\ Antti Karttunen, May 25 2017

a(n) = A000005(n) - A049419(n) for n >= 2.
a(1) = 1, a(p) = 1, a(p*q) = 3, a(p*q*...*z) = 2^k - 1, where the indices are p=primes (A000040), p*q = product of two distinct primes (A006881), and generally p*q*...*z = product of k (k > 0) distinct primes (A120944).
a(p^k) = k + 1 - A000005(k), where p are primes (A000040), p^k are prime powers A000961 (n>1), k = natural numbers (A000027).
a(p^q) = q - 1, where p and q are primes (A000040), and p^q = prime powers of primes (A053810).
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*A001620 - A327837 - 1). - Amiram Eldar, Feb 03 2025

Edited by R. J. Mathar, May 08 2009

A368980 The number of exponential divisors of n that are squares (A000290).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0

Offset: 1

Amiram Eldar, Jan 11 2024

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

Cf. A000005, A000037, A000290, A049419, A183063, A322791, A327837, A368979.

Similar sequences: A046951, A056624, A358260, A368978.

f[p_, e_] := If[OddQ[e], 0, DivisorSigma[0, e/2]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x%2, 0, numdiv(x/2)), factor(n)[, 2]));

a(n^2) = A049419(n). [corrected by Ridouane Oudra, Nov 19 2024]
Multiplicative with a(p^e) = A183063(e), or equivalently, a(p^e) = 0 if e is odd, and A000005(e/2) if e is even.
a(n) >= 0, with equality if and only if n is not a square number (A000037).
a(n) <= A049419(n), with equality if and only if n = 1.
Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = 1.602317... (A327837).

A379519 Numerators of the partial alternating sums of the reciprocals of the unitary totient function (A047994).

Original entry on oeis.org

1, 0, 1, 1, 5, -1, 1, -5, 11, -31, -71, -211, -47, -281, -22, -29, -359, -569, -1427, -1847, -1427, -1931, -18721, -22681, -20371, -24991, -297163, -37467, -34607, -44617, -125843, -4141373, -3769001, -2117233, -327013, -2117233, -6041389, -6662009, -774568, -3297757

Offset: 1

Amiram Eldar, Dec 24 2024

			Fractions begin with 1, 0, 1/2, 1/6, 5/12, -1/12, 1/12, -5/84, 11/168, -31/168, -71/840, -211/840, ...

Amiram Eldar, Table of n, a(n) for n = 1..1000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.10, pp. 30-31.

Cf. A047994, A065442, A177754, A327837, A370899, A379517, A379520 (denominators).

uphi[n_] := Times @@ (-1 + Power @@@ FactorInteger[n]); uphi[1] = 1; Numerator[Accumulate[Table[(-1)^(n+1)/uphi[n], {n, 1, 50}]]]

uphi(n) = {my(f = factor(n)); prod(i = 1, #f~, -1 + f[i, 1]^f[i, 2]);}
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / uphi(k); print1(numerator(s), ", "))};

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A047994(k)).

a(n)/A379520(n) = T * log(n) + U + O(log(n)^(5/3) / n^u), where u > 0, T = A327837 * (2/(A065442 + 1) - 1), and U is a constant.

cp's OEIS Frontend

A049419 a(1) = 1; for n > 1, a(n) = number of exponential divisors of n.

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A361967 Number of numbers k such that uphi(k) = n, where uphi is the unitary totient function (A047994).

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A145353 Sum of the number of e-divisors of all numbers from 1 up to 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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A327838 Decimal expansion of the asymptotic mean of the exponential totient function (A072911).

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A379517 Numerators of the partial sums of the reciprocals of the unitary totient function (A047994).

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A160097 Number of non-exponential divisors of n.

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