This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A379519 #7 Dec 24 2024 07:26:25
%S A379519 1,0,1,1,5,-1,1,-5,11,-31,-71,-211,-47,-281,-22,-29,-359,-569,-1427,
%T A379519 -1847,-1427,-1931,-18721,-22681,-20371,-24991,-297163,-37467,-34607,
%U A379519 -44617,-125843,-4141373,-3769001,-2117233,-327013,-2117233,-6041389,-6662009,-774568,-3297757
%N A379519 Numerators of the partial alternating sums of the reciprocals of the unitary totient function (A047994).
%H A379519 Amiram Eldar, <a href="/A379519/b379519.txt">Table of n, a(n) for n = 1..1000</a>
%H A379519 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 A379519 a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A047994(k)).
%F A379519 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 A379519 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 A379519 uphi[n_] := Times @@ (-1 + Power @@@ FactorInteger[n]); uphi[1] = 1; Numerator[Accumulate[Table[(-1)^(n+1)/uphi[n], {n, 1, 50}]]]
%o A379519 (PARI) uphi(n) = {my(f = factor(n)); prod(i = 1, #f~, -1 + f[i, 1]^f[i, 2]);}
%o A379519 list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / uphi(k); print1(numerator(s), ", "))};
%Y A379519 Cf. A047994, A065442, A177754, A327837, A370899, A379517, A379520 (denominators).
%K A379519 sign,easy,frac
%O A379519 1,5
%A A379519 _Amiram Eldar_, Dec 24 2024