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
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
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
CROSSREFS
KEYWORD
nonn,mult,nice
AUTHOR
EXTENSIONS
More terms from Jud McCranie, May 29 2000
STATUS
approved