A166234 The inverse of the constant 1 function under the exponential convolution (also called the exponential Möbius function).

1, 1, 1, -1, 1, 1, 1, -1, -1, 1, 1, -1, 1, 1, 1, 0, 1, -1, 1, -1, 1, 1, 1, -1, -1, 1, -1, -1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, -1, -1, 1, 1, 0, -1, -1, 1, -1, 1, -1, 1, -1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, -1, -1, 1, 1, 1, 0, 0, 1, 1, -1, 1, 1, 1, -1, 1, -1, 1

Offset

1,1

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Xiaodong Cao and Wenguang Zahi, Some arithmetic functions involving exponential divisors, Journal of Integer Sequences, Vol. 13 (2010), Article 10.3.7.

Andrew V. Lelechenko, Exponential and infinitary divisors, Ukrainian Mathematical Journal, Vol. 68, No. 8 (2017), pp. 1222-1237; arXiv preprint, arXiv:1405.7597 [math.NT], 2014, function mu^(E)(n).

M. V. Subbarao, On some arithmetic convolutions, in: A. A. Gioia and D. L. Goldsmith (eds.), The Theory of Arithmetic Functions, Lecture Notes in Mathematics No. 251, Springer, 1972, pp. 247-271; alternative link.

László Tóth, On certain arithmetic functions involving exponential divisors, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 24 (2004), pp. 285-296; arXiv preprint, arXiv:math/0610274 [math.NT], 2006-2009.

László Tóth, On certain arithmetic functions involving exponential divisors, II, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 27 (2007), pp. 155-166; arXiv preprint, arXiv:0708.3557 [math.NT], 2007-2009.

Formula

Multiplicative, a(p^e) = mu(e) for any prime power p^e (e>=1), where mu is the Möbius function A008683.

a(A130897(n)) = 0; a(A209061(n)) <> 0. - Reinhard Zumkeller, Mar 13 2012

Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = Product_{p prime} (1 + Sum_{k>=2} (mu(k) - mu(k-1))/p^k) = 0.3609447238... (Tóth, 2007). - Amiram Eldar, Nov 08 2020

Maple

A166234 := proc(n)
    local a, p;
    a := 1;
    if n =1 then
        ;
    else
        for p in ifactors(n)[2] do
                    a := a*numtheory[mobius](op(2, p)) ;
        end do:
    end if;
    a ;
end proc:# R. J. Mathar, Nov 30 2016

Mathematica

a[n_] := Times @@ MoebiusMu /@ FactorInteger[n][[All, 2]];
Array[a, 100] (* Jean-François Alcover, Nov 16 2017 *)

Prog

(Haskell)
a166234 = product . map (a008683 . fromIntegral) . a124010_row
-- Reinhard Zumkeller, Mar 13 2012
(PARI) a(n)=factorback(apply(moebius, factor(n)[, 2])) \\ Charles R Greathouse IV, Sep 02 2015

Crossrefs

Cf. A008683, A049419, A051377, A124010, A209802 (partial sums).

Sequence in context: A053865 A189022 A370598 * A074481 A015420 A015522

Adjacent sequences: A166231 A166232 A166233 * A166235 A166236 A166237

Keyword

mult,sign

Author

Laszlo Toth, Oct 09 2009

Status

approved