cp's OEIS Frontend

A209802 Partial sums of exponential Möbius function, A166234.

%I A209802 #29 May 30 2025 03:32:08
%S A209802 1,2,3,2,3,4,5,4,3,4,5,4,5,6,7,7,8,7,8,7,8,9,10,9,8,9,8,7,8,9,10,9,10,
%T A209802 11,12,13,14,15,16,15,16,17,18,17,16,17,18,18,17,16,17,16,17,16,17,16,
%U A209802 17,18,19,18,19,20,19,20,21,22,23,22,23,24,25,26
%N A209802 Partial sums of exponential Möbius function, A166234.
%C A209802 Analog of Mertens's function, A002321; conjecture: a(n) > 0.
%C A209802 Values of a(10^n) at n = 1, 2, 3, ...: 4, 34, 355, 3610, 36116, 360967, 3609566, 36094237, .... - _Charles R Greathouse IV_, Sep 02 2015
%H A209802 Reinhard Zumkeller, <a href="/A209802/b209802.txt">Table of n, a(n) for n = 1..10000</a>
%H A209802 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 A209802 a(n) ~ c * n, where c = Product_{p prime} (1 + Sum_{k>=2} (mu(k) - mu(k-1))/p^k) = 0.3609447238... (Tóth, 2007). - _Amiram Eldar_, Nov 08 2020
%t A209802 f[p_, e_] := MoebiusMu[e]; em[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate @ Array[em, 100] (* _Amiram Eldar_, Nov 08 2020 *)
%o A209802 (Haskell)
%o A209802 a209802 n = a209802_list !! (n-1)
%o A209802 a209802_list = scanl1 (+) a166234_list
%o A209802 (PARI) first(n)=my(s); vector(n,k, s+=factorback(apply(moebius, factor(k)[,2]))) \\ _Charles R Greathouse IV_, Sep 02 2015
%o A209802 (PARI) a(n)=sum(k=1,n,factorback(apply(moebius, factor(k)[,2]))) \\ _Charles R Greathouse IV_, Sep 02 2015
%Y A209802 Cf. A002321, A166234.
%K A209802 nonn
%O A209802 1,2
%A A209802 _Reinhard Zumkeller_, Mar 13 2012