A379616 - OEIS

%I #7 Dec 28 2024 09:11:30

%S 1,3,12,60,20,30,120,40,40,360,360,72,504,126,504,1512,1512,7560,1512,

%T 7560,30240,30240,30240,30240,393120,393120,393120,393120,393120,

%U 393120,196560,28080,14040,4680,9360,46800,889200,889200,6224400,6224400,889200,1778400

%N Denominators of the partial sums of the reciprocals of the sum of bi-unitary divisors function (A188999).

%H Amiram Eldar, <a href="/A379616/b379616.txt">Table of n, a(n) for n = 1..1000</a>

%H V. Sitaramaiah and M. V. Subbarao, <a href="https://informaticsjournals.co.in/index.php/jims/article/view/21936">Asymptotic formulae for sums of reciprocals of some multiplicative functions</a>, J. Indian Math. Soc., Vol. 57 (1991), pp. 153-167.

%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 4.13, p. 34.

%F a(n) = denominator(Sum_{k=1..n} 1/A188999(k)).

%t f[p_, e_] := (p^(e+1) - 1)/(p - 1) - If[OddQ[e], 0, p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ f @@@ FactorInteger[n]; Denominator[Accumulate[Table[1/bsigma[n], {n, 1, 50}]]]

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

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

%Y Cf. A188999, A307159, A370904, A379615 (numerators), A379618.

%K nonn,easy,frac,new

%O 1,2

%A _Amiram Eldar_, Dec 27 2024