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

%I A166234 #49 May 30 2025 03:15:02
%S A166234 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,
%T A166234 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,
%U A166234 -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
%N A166234 The inverse of the constant 1 function under the exponential convolution (also called the exponential Möbius function).
%H A166234 Reinhard Zumkeller, <a href="/A166234/b166234.txt">Table of n, a(n) for n = 1..10000</a>
%H A166234 Xiaodong Cao and Wenguang Zahi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Cao/cao4.html">Some arithmetic functions involving exponential divisors</a>, Journal of Integer Sequences, Vol. 13 (2010), Article 10.3.7.
%H A166234 Andrew V. Lelechenko, <a href="https://doi.org/10.1007/s11253-017-1289-7">Exponential and infinitary divisors</a>, Ukrainian Mathematical Journal, Vol. 68, No. 8 (2017), pp. 1222-1237; <a href="http://arxiv.org/abs/1405.7597">arXiv preprint</a>, arXiv:1405.7597 [math.NT], 2014, function mu^(E)(n).
%H A166234 M. V. Subbarao, <a href="http://dx.doi.org/10.1007/BFb0058796">On some arithmetic convolutions</a>, 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; <a href="https://web.archive.org/web/20110910073428/http://www.math.ualberta.ca/~subbarao/documents/Subbarao7.pdf">alternative link</a>.
%H A166234 László Tóth, <a href="https://ac.inf.elte.hu/Vol_024_2004/doi/285_24.pdf">On certain arithmetic functions involving exponential divisors</a>, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 24 (2004), pp. 285-296; <a href="https://arxiv.org/abs/math/0610274">arXiv preprint</a>, arXiv:math/0610274 [math.NT], 2006-2009.
%H A166234 László Tóth, <a href="https://ac.inf.elte.hu/Vol_027_2007/doi/155_27.pdf">On certain arithmetic functions involving exponential divisors, II</a>, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 27 (2007), pp. 155-166; <a href="https://arxiv.org/abs/0708.3557">arXiv preprint</a>, arXiv:0708.3557 [math.NT], 2007-2009.
%F A166234 Multiplicative, a(p^e) = mu(e) for any prime power p^e (e>=1), where mu is the Möbius function A008683.
%F A166234 a(A130897(n)) = 0; a(A209061(n)) <> 0. - _Reinhard Zumkeller_, Mar 13 2012
%F A166234 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
%p A166234 A166234 := proc(n)
%p A166234     local a,p;
%p A166234     a := 1;
%p A166234     if n =1 then
%p A166234         ;
%p A166234     else
%p A166234         for p in ifactors(n)[2] do
%p A166234                     a := a*numtheory[mobius](op(2,p)) ;
%p A166234         end do:
%p A166234     end if;
%p A166234     a ;
%p A166234 end proc:# _R. J. Mathar_, Nov 30 2016
%t A166234 a[n_] := Times @@ MoebiusMu /@ FactorInteger[n][[All, 2]];
%t A166234 Array[a, 100] (* _Jean-François Alcover_, Nov 16 2017 *)
%o A166234 (Haskell)
%o A166234 a166234 = product . map (a008683 . fromIntegral) . a124010_row
%o A166234 -- _Reinhard Zumkeller_, Mar 13 2012
%o A166234 (PARI) a(n)=factorback(apply(moebius, factor(n)[,2])) \\ _Charles R Greathouse IV_, Sep 02 2015
%Y A166234 Cf. A008683, A049419, A051377, A124010, A209802 (partial sums).
%K A166234 mult,sign
%O A166234 1,1
%A A166234 _Laszlo Toth_, Oct 09 2009