A379617 - OEIS

%I #7 Dec 28 2024 09:10:55

%S 1,2,11,43,53,4,37,103,23,65,71,337,2539,1217,2539,7337,7757,1501,

%T 7883,7631,31469,30629,31889,6277,84625,82753,423593,82753,426869,

%U 421409,216847,213727,108911,11899,24253,119081,2317139,760853,773203,6889667,7037867,13946059

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

%H Amiram Eldar, <a href="/A379617/b379617.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 4.13, p. 34.

%F a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A188999(k)).

%F a(n)/A379618(n) = A * log(n) + B + O(log(n)^(14/3) * log(log(n))^(4/3) * n^c), where c = log(9/10)/log(2) = -0.152003..., and A and B are constants.

%e Fractions begin with 1, 2/3, 11/12, 43/60, 53/60, 4/5, 37/40, 103/120, 23/24, 65/72, 71/72, 337/360, ...

%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]; Numerator[Accumulate[Table[(-1)^(n+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)^(k+1) / bsigma(k); print1(numerator(s), ", "))};

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

%K nonn,easy,frac

%O 1,2

%A _Amiram Eldar_, Dec 27 2024