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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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

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

From Amiram Eldar, Dec 20 2025: (Start)

The inverse Moebius transform is not A124315, since the Moebius transform of A124315 is A046951, which differs from this sequence at n = 32, 96, 128, 160, 224, 243, 256, 288, ... .

The Dirichlet inverse is the multiplicative function f(n) with f(p^e) = -A159933(e). (End)

LINKS

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.

FORMULA

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

Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 + Sum_{k>=5} (tau(k) - tau(k-1) - tau(k-2) + tau(k-3))/p^(k*s)), where tau = A000005 (Wu, 1995). - Amiram Eldar, Dec 20 2025

EXAMPLE

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}

MAPLE

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

MATHEMATICA

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 *)

PROG

(Haskell)

a049419 = product . map (a000005 . fromIntegral) . a124010_row

-- Reinhard Zumkeller, Mar 13 2012

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

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

(Python)

from math import prod

from sympy import divisor_count, factorint

def A049419(n): return prod(divisor_count(e) for e in factorint(n).values()) # Chai Wah Wu, Sep 26 2025

CROSSREFS

Row lengths of A322791.