A209802 - OEIS

%I #26 Nov 08 2020 03:30:59

%S 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 11,12,13,14,15,16,15,16,17,18,17,16,17,18,18,17,16,17,16,17,16,17,16,

%U 17,18,19,18,19,20,19,20,21,22,23,22,23,24,25,26

%N Partial sums of exponential Möbius function, A166234.

%C Analog of Mertens's function, A002321; conjecture: a(n) > 0.

%C 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 Reinhard Zumkeller, <a href="/A209802/b209802.txt">Table of n, a(n) for n = 1..10000</a>

%H László Tóth, <a href="http://ac.inf.elte.hu/Vol_027_2007/155.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 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 f[p_, e_] := MoebiusMu[e]; em[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate @ Array[em, 100] (* _Amiram Eldar_, Nov 08 2020 *)

%o (Haskell)

%o a209802 n = a209802_list !! (n-1)

%o a209802_list = scanl1 (+) a166234_list

%o (PARI) first(n)=my(s); vector(n,k, s+=factorback(apply(moebius, factor(k)[,2]))) \\ _Charles R Greathouse IV_, Sep 02 2015

%o (PARI) a(n)=sum(k=1,n,factorback(apply(moebius, factor(k)[,2]))) \\ _Charles R Greathouse IV_, Sep 02 2015

%Y Cf. A002321, A166234.

%K nonn

%O 1,2

%A _Reinhard Zumkeller_, Mar 13 2012