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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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48
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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
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OFFSET
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1,4
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COMMENTS
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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
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LINKS
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Eric Weisstein's World of Mathematics, e-Divisor.
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FORMULA
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EXAMPLE
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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}
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MAPLE
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local a;
a := 1 ;
for pf in ifactors(n)[2] do
a := a*numtheory[tau](op(2, pf)) ;
end do:
a ;
end proc:
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
a049419 = product . map (a000005 . fromIntegral) . a124010_row
(PARI) a(n) = vecprod(apply(numdiv, factor(n)[, 2])); \\ Amiram Eldar, Mar 27 2023
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CROSSREFS
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KEYWORD
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nonn,mult,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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