%I #8 Dec 24 2023 02:43:15

%S 1,0,-1,-2,-3,-2,-3,-2,-3,-2,-3,-2,-3,-2,-1,-2,-3,-2,-3,-2,-1,0,-1,-2,

%T -3,-2,-1,0,-1,-2,-3,-2,-1,0,1,2,1,2,3,2,1,0,-1,0,1,2,1,2,1,2,3,4,3,2,

%U 3,2,3,4,3,2,1,2,3,4,5,4,3,4,5,4,3,2,1,2,3,4,5

%N Infinitary version of Mertens's function: a(n) = Sum_{k=1..n} A064179(k).

%H Amiram Eldar, <a href="/A368405/b368405.txt">Table of n, a(n) for n = 1..10000</a>

%H Rasa Steuding, Jörn Steuding, and László Tóth, <a href="https://doi.org/10.1007/s12215-011-0022-x">A modified Möbius mu-function</a>, Rendiconti del Circolo Matematico di Palermo, Vol. 60 (2011), pp. 13-21; <a href="https://arxiv.org/abs/1109.4242">arXiv preprint</a>, arXiv:1109.4242 [math.NT], 2011.

%t f[p_, e_] := (-1)^DigitCount[e, 2, 1]; imu[1] = 1; imu[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Array[imu, 100]]

%o (PARI) imu(n) = vecprod(apply(x -> (-1)^hammingweight(x), factor(n)[, 2]));

%o lista(nmax) = {my(s = 0); for(k = 1, nmax, s+ = imu(k); print1(s, ", "));}

%Y Partial sums of A064179.

%Y Similar sequences: A002321, A174863 (unitary), A209802 (exponential).

%K sign,easy

%O 1,4

%A _Amiram Eldar_, Dec 23 2023