A379369 - OEIS

%I #6 Dec 22 2024 16:54:09

%S 1,1,5,1,8,11,107,1,12,17,257,193,3664,5183,479,-261,-3436,-37633,

%T -612925,-2017297,-1786352,-4013599,-82613087,-19965872,-12529443,

%U -27919051,-9392863,-12664034,-255710551,-242359181,-5356570201,-19391659278,-55136182529,-116171203003

%N Numerators of the partial alternating sums of the reciprocals of the squarefree kernel function (A007947).

%H Amiram Eldar, <a href="/A379369/b379369.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 4.6, pp. 24-26.

%F a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A007947(k)).

%F a(n)/A379370(n) ~ -A379367(n)/A379368(n).

%e Fractions begin with 1, 1/2, 5/6, 1/3, 8/15, 11/30, 107/210, 1/105, 12/35, 17/70, 257/770, 193/1155, ...

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

%o (PARI) rad(n) = vecprod(factor(n)[, 1]);

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

%Y Cf. A007947, A073355, A370896, A379367, A379368, A379370 (denominators).

%K sign,easy,frac

%O 1,3

%A _Amiram Eldar_, Dec 21 2024