A379363 Numerators of the partial sums of the reciprocals of Pillai's arithmetical function (A018804).

1, 4, 23, 199, 637, 661, 8953, 9187, 65869, 201247, 205927, 26048, 132697, 134272, 135637, 2190667, 24424937, 3513791, 131554667, 132348317, 133227437, 938941259, 947830139, 190366027, 2947643, 74101331, 223443593, 2916305159, 55809797621, 55978686341, 3437499844001

Offset

1,2

Links

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

László Tóth, A survey of gcd-sum functions, Journal of Integer Sequences, Vol. 13 (2010), Article 10.8.1. See pp. 18-19.

László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.5, pp. 23-24.

Shiqin Chen and Wenguang Zhai, Reciprocals of the Gcd-Sum Functions, Journal of Integer Sequences, Vol. 14 (2011), Article 11.8.3.

Formula

a(n) = numerator(Sum_{k=1..n} 1/A018804(k)).

a(n)/A379364(n) = Sum_{j=0..N} K_j/log(n)^(j-1/2) + O(1/log(n)^(N+1/2)), for any integer N >= 1, where K_j are constants, and in particular K_0 = (2/sqrt(Pi)) * Product_{p prime} (sqrt(1-1/p) * Sum_{k>=1} 1/A018804(p^k)) = 1.30088863073811791549... .

Example

Fractions begin with 1, 4/3, 23/15, 199/120, 637/360, 661/360, 8953/4680, 9187/4680, 65869/32760, 201247/98280, 205927/98280, 26048/12285, ...

Mathematica

f[p_, e_] := (e*(p-1)/p + 1)*p^e; pillai[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[Accumulate[Table[1/pillai[n], {n, 1, 50}]]]

Prog

(PARI) pillai(n) = {my(f=factor(n)); prod(i=1, #f~, (f[i, 2]*(f[i, 1]-1)/f[i, 1] + 1)*f[i, 1]^f[i, 2]); }
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / pillai(k); print1(numerator(s), ", "))};

Crossrefs

Cf. A018804, A272718, A370895, A379364 (denominators), A379365.

Sequence in context: A222454 A336948 A099869 * A369194 A365353 A265677

Adjacent sequences: A379360 A379361 A379362 * A379364 A379365 A379366

Keyword

nonn,easy,frac

Author

Amiram Eldar, Dec 21 2024

Status

approved