%I #73 Mar 27 2023 03:47:06

%S 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,

%T 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,

%U 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

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

%C 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.

%C Wu gives a complicated Dirichlet g.f.

%C a(1) = 1 by convention. This is also required for a function to be multiplicative. - _N. J. A. Sloane_, Mar 03 2009

%C The inverse Moebius transform seems to be in A124315. The Dirichlet inverse appears to be related to A166234. - _R. J. Mathar_, Jul 14 2014

%H Reinhard Zumkeller, <a href="/A049419/b049419.txt">Table of n, a(n) for n = 1..10000</a>

%H Andrew V. Lelechenko, <a href="http://arxiv.org/abs/1405.7597">Exponential and infinitary divisors</a>, arXiv:1405.7597 [math.NT], 2014, sequence tau^(e).

%H David Moews, <a href="http://djm.cc/aliquot-database/aliquot-database.uhtml">A database of aliquot cycles</a>.

%H J. O. M. Pedersen, <a href="http://62.198.248.44/aliquot/tables.htm">Tables of Aliquot Cycles</a>.

%H J. O. M. Pedersen, <a href="http://web.archive.org/web/20140502102524/http://amicable.homepage.dk/tables.htm">Tables of Aliquot Cycles</a>. [Via Internet Archive Wayback-Machine]

%H J. O. M. Pedersen, <a href="/A063990/a063990.pdf">Tables of Aliquot Cycles</a>. [Cached copy, pdf file only]

%H László Tóth and Nicuşor Minculete, <a href="http://arxiv.org/abs/0910.2798">Exponential unitary divisors</a>, arXiv:0910.2798 [math.NT], 2009.

%H Tim Trudgian, <a href="http://arxiv.org/abs/1312.4615">The sum of the unitary divisor function</a>, arXiv:1312.4615 [math.NT], 2013-2014, Section 3.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/e-Divisor.html">e-Divisor</a>.

%H Jie Wu, <a href="http://dx.doi.org/10.5802/jtnb.136">Problème de diviseurs exponentiels et entiers exponentiellement sans facteur carré</a>, J. Theor. Nombr. Bordeaux 7 (1) (1995) 133-141.

%F Multiplicative with a(p^e) = tau(e). - _Vladeta Jovovic_, Jul 23 2001

%F Sum_{k=1..n} a(k) ~ A327837 * n. - _Vaclav Kotesovec_, Feb 27 2023

%e a(8)=2 because 2 and 2^3 are e-divisors of 8.

%e The sets of e-divisors start as:

%e 1:{1}

%e 2:{2}

%e 3:{3}

%e 4:{2, 4}

%e 5:{5}

%e 6:{6}

%e 7:{7}

%e 8:{2, 8}

%e 9:{3, 9}

%e 10:{10}

%e 11:{11}

%e 12:{6, 12}

%e 13:{13}

%e 14:{14}

%e 15:{15}

%e 16:{2, 4, 16}

%e 17:{17}

%e 18:{6, 18}

%e 19:{19}

%e 20:{10, 20}

%e 21:{21}

%e 22:{22}

%e 23:{23}

%e 24:{6, 24}

%p A049419 := proc(n)

%p local a;

%p a := 1 ;

%p for pf in ifactors(n)[2] do

%p a := a*numtheory[tau](op(2,pf)) ;

%p end do:

%p a ;

%p end proc:

%p seq(A049419(n),n=1..20) ; # _R. J. Mathar_, Jul 14 2014

%t 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_ *)

%o (Haskell)

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

%o -- _Reinhard Zumkeller_, Mar 13 2012

%o (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

%o (PARI) a(n) = vecprod(apply(numdiv, factor(n)[,2])); \\ _Amiram Eldar_, Mar 27 2023

%Y Row lengths of A322791.

%Y Cf. A049599, A061389, A051377 (sum of e-divisors).

%Y Partial sums are in A099593.

%Y Cf. A124010, A000005, A049599, A072911, A124315, A166234, A327837, A361012.

%K nonn,mult,nice

%O 1,4

%A _Yasutoshi Kohmoto_

%E More terms from _Jud McCranie_, May 29 2000