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

%I A049419 #74 Feb 16 2025 08:32:40
%S A049419 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 A049419 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 A049419 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 A049419 a(1) = 1; for n > 1, a(n) = number of exponential divisors of n.
%C A049419 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 A049419 Wu gives a complicated Dirichlet g.f.
%C A049419 a(1) = 1 by convention. This is also required for a function to be multiplicative. - _N. J. A. Sloane_, Mar 03 2009
%C A049419 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 A049419 Reinhard Zumkeller, <a href="/A049419/b049419.txt">Table of n, a(n) for n = 1..10000</a>
%H A049419 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 A049419 David Moews, <a href="http://djm.cc/aliquot-database/aliquot-database.uhtml">A database of aliquot cycles</a>.
%H A049419 J. O. M. Pedersen, <a href="http://62.198.248.44/aliquot/tables.htm">Tables of Aliquot Cycles</a>.
%H A049419 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 A049419 J. O. M. Pedersen, <a href="/A063990/a063990.pdf">Tables of Aliquot Cycles</a>. [Cached copy, pdf file only]
%H A049419 László Tóth and Nicuşor Minculete, <a href="http://arxiv.org/abs/0910.2798">Exponential unitary divisors</a>, arXiv:0910.2798 [math.NT], 2009.
%H A049419 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 A049419 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/e-Divisor.html">e-Divisor</a>.
%H A049419 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 A049419 Multiplicative with a(p^e) = tau(e). - _Vladeta Jovovic_, Jul 23 2001
%F A049419 Sum_{k=1..n} a(k) ~ A327837 * n. - _Vaclav Kotesovec_, Feb 27 2023
%e A049419 a(8)=2 because 2 and 2^3 are e-divisors of 8.
%e A049419 The sets of e-divisors start as:
%e A049419   1:{1}
%e A049419   2:{2}
%e A049419   3:{3}
%e A049419   4:{2, 4}
%e A049419   5:{5}
%e A049419   6:{6}
%e A049419   7:{7}
%e A049419   8:{2, 8}
%e A049419   9:{3, 9}
%e A049419   10:{10}
%e A049419   11:{11}
%e A049419   12:{6, 12}
%e A049419   13:{13}
%e A049419   14:{14}
%e A049419   15:{15}
%e A049419   16:{2, 4, 16}
%e A049419   17:{17}
%e A049419   18:{6, 18}
%e A049419   19:{19}
%e A049419   20:{10, 20}
%e A049419   21:{21}
%e A049419   22:{22}
%e A049419   23:{23}
%e A049419   24:{6, 24}
%p A049419 A049419 := proc(n)
%p A049419     local a;
%p A049419     a := 1 ;
%p A049419     for pf in ifactors(n)[2] do
%p A049419         a := a*numtheory[tau](op(2,pf)) ;
%p A049419     end do:
%p A049419     a ;
%p A049419 end proc:
%p A049419 seq(A049419(n),n=1..20) ; # _R. J. Mathar_, Jul 14 2014
%t A049419 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 A049419 (Haskell)
%o A049419 a049419 = product . map (a000005 . fromIntegral) . a124010_row
%o A049419 -- _Reinhard Zumkeller_, Mar 13 2012
%o A049419 (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 A049419 (PARI) a(n) = vecprod(apply(numdiv, factor(n)[,2])); \\ _Amiram Eldar_, Mar 27 2023
%Y A049419 Row lengths of A322791.
%Y A049419 Cf. A049599, A061389, A051377 (sum of e-divisors).
%Y A049419 Partial sums are in A099593.
%Y A049419 Cf. A124010, A000005, A049599, A072911, A124315, A166234, A327837, A361012.
%K A049419 nonn,mult,nice
%O A049419 1,4
%A A049419 _Yasutoshi Kohmoto_
%E A049419 More terms from _Jud McCranie_, May 29 2000