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A049419
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a(1) = 1; for n > 1, a(n) = number of exponential divisors of n.
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13
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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; internal format)
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OFFSET
| 1,4
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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
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LINKS
| J. O. M. Pedersen, Tables of Aliquot Cycles
Laszlo Toth, Nicusor Minculete, Exponential unitary divisors, arXiv:0910.2798 [math.NT]
Eric Weisstein's World of Mathematics, Definition
J. Wu, Probleme de diviseurs exponentiels..., J. Theor. Nombr. Bordeaux 7 (1) (1995) 133-141.
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FORMULA
| Multiplicative with a(p^e) = tau(e). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 23 2001
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EXAMPLE
| a(8)=2 because 2 and 2^3 are e-divisors of 8.
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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}] (* From Jean-François Alcover, Jan 30 2012, after Vladeta Jovovic *)
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CROSSREFS
| Cf. A049599, A061389, A051377.
Partial sums are in A099593.
Sequence in context: A085424 A088737 A096309 * A046951 A159631 A050377
Adjacent sequences: A049416 A049417 A049418 * A049420 A049421 A049422
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KEYWORD
| nonn,mult,nice
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AUTHOR
| Yasutoshi Kohmoto (zbi74583(AT)boat.zero.ad.jp)
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EXTENSIONS
| More terms from Jud McCranie (JudMcCranie(AT)ugaalum.uga.edu), May 29 2000
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