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 A049419 a(1) = 1; for n > 1, a(n) = number of exponential divisors of n. 28
 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 LINKS Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 A. 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 [Broken link] 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] Laszlo Toth, Nicuşor Minculete, Exponential unitary divisors, arXiv:0910.2798 [math.NT], 2009. T. Trudgian, The sum of the unitary divisor function, arXiv:1312.4615 [math.NT], Section 3. Eric Weisstein's World of Mathematics, e-Divisor J. 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 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 CROSSREFS Row lengths of A322791. Cf. A049599, A061389, A051377 (sum of e-divisors). Partial sums are in A099593. Cf. A124010, A000005, A049599, A072911. Sequence in context: A321455 A096309 A185102 * A299090 A046951 A159631 Adjacent sequences:  A049416 A049417 A049418 * A049420 A049421 A049422 KEYWORD nonn,mult,nice AUTHOR EXTENSIONS More terms from Jud McCranie, May 29 2000 STATUS approved

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Last modified April 10 16:05 EDT 2021. Contains 342845 sequences. (Running on oeis4.)