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

Author

Laszlo Toth, Oct 09 2009

Keywords

• mult
• sign

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.

Crossrefs

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

Programs

Haskell

a166234 = product . map (a008683 . fromIntegral) . a124010_row
-- Reinhard Zumkeller, Mar 13 2012

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 *)

PARI

a(n)=factorback(apply(moebius, factor(n)[,2])) \\ Charles R Greathouse IV, Sep 02 2015

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