A379582 - OEIS

%I #6 Dec 26 2024 20:02:49

%S 1,2,6,6,30,15,105,105,105,210,2310,2310,30030,15015,1001,1001,17017,

%T 34034,646646,3233230,9699690,4849845,111546435,37182145,37182145,

%U 74364290,74364290,74364290,2156564410,3234846615,100280245065,100280245065,100280245065,200560490130

%N Denominators of the partial alternating sums of the reciprocals of the powerfree part function (A055231).

%H Amiram Eldar, <a href="/A379582/b379582.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 4.11, pp. 31-32.

%F a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/A055231(k)).

%t f[p_, e_] := If[e==1, p, 1]; powfree[n_] := Times @@ f @@@ FactorInteger[n]; Denominator[Accumulate[Table[(-1)^(n+1)/powfree[n], {n, 1, 50}]]]

%o (PARI) powfree(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] == 1, f[i, 1], 1)); }

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

%Y Cf. A055231, A370900, A370901, A379580, A379581 (numerators).

%K nonn,easy,frac

%O 1,2

%A _Amiram Eldar_, Dec 26 2024