A379615 Numerators of the partial sums of the reciprocals of the sum of bi-unitary divisors function (A188999).

1, 4, 19, 107, 39, 61, 259, 89, 93, 857, 887, 181, 1303, 331, 1345, 4091, 4175, 21127, 4301, 21757, 87973, 88813, 90073, 90577, 1192621, 1201981, 1211809, 1221637, 1234741, 1240201, 626243, 89909, 45247, 15169, 30533, 153601, 2941819, 2956639, 20807623, 20876783

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..1000

V. Sitaramaiah and M. V. Subbarao, Asymptotic formulae for sums of reciprocals of some multiplicative functions, J. Indian Math. Soc., Vol. 57 (1991), pp. 153-167.

László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.13, p. 34.

Formula

a(n) = numerator(Sum_{k=1..n} 1/A188999(k)).

a(n)/A379616(n) = A * log(n) + B + O(log(n)^(14/3) * log(log(n))^(4/3) / n), where A and B are constants.

Example

Fractions begin with 1, 4/3, 19/12, 107/60, 39/20, 61/30, 259/120, 89/40, 93/40, 857/360, 887/360, 181/72, ...

Mathematica

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/bsigma[n], {n, 1, 50}]]]

Prog

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

Crossrefs

Cf. A188999, A307159, A370904, A379616 (denominators), A379617.

Sequence in context: A369109 A082030 A348802 * A379513 A052751 A367284

Adjacent sequences: A379612 A379613 A379614 * A379616 A379617 A379618

Keyword

nonn,easy,frac

Author

Amiram Eldar, Dec 27 2024

Status

approved