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A049419 a(1) = 1; for n > 1, a(n) = number of exponential divisors of n. 22
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

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

Yasutoshi Kohmoto

EXTENSIONS

More terms from Jud McCranie, May 29 2000

STATUS

approved

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Last modified January 18 16:40 EST 2019. Contains 319271 sequences. (Running on oeis4.)