A379519 - OEIS

%I #7 Dec 24 2024 07:26:25

%S 1,0,1,1,5,-1,1,-5,11,-31,-71,-211,-47,-281,-22,-29,-359,-569,-1427,

%T -1847,-1427,-1931,-18721,-22681,-20371,-24991,-297163,-37467,-34607,

%U -44617,-125843,-4141373,-3769001,-2117233,-327013,-2117233,-6041389,-6662009,-774568,-3297757

%N Numerators of the partial alternating sums of the reciprocals of the unitary totient function (A047994).

%H Amiram Eldar, <a href="/A379519/b379519.txt">Table of n, a(n) for n = 1..1000</a>

%H László Tóth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Toth/toth25.html">Alternating Sums Concerning Multiplicative Arithmetic Functions</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.10, pp. 30-31.

%F a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A047994(k)).

%F a(n)/A379520(n) = T * log(n) + U + O(log(n)^(5/3) / n^u), where u > 0, T = A327837 * (2/(A065442 + 1) - 1), and U is a constant.

%e Fractions begin with 1, 0, 1/2, 1/6, 5/12, -1/12, 1/12, -5/84, 11/168, -31/168, -71/840, -211/840, ...

%t uphi[n_] := Times @@ (-1 + Power @@@ FactorInteger[n]); uphi[1] = 1; Numerator[Accumulate[Table[(-1)^(n+1)/uphi[n], {n, 1, 50}]]]

%o (PARI) uphi(n) = {my(f = factor(n)); prod(i = 1, #f~, -1 + f[i, 1]^f[i, 2]);}

%o list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / uphi(k); print1(numerator(s), ", "))};

%Y Cf. A047994, A065442, A177754, A327837, A370899, A379517, A379520 (denominators).

%K sign,easy,frac

%O 1,5

%A _Amiram Eldar_, Dec 24 2024