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

Author

Amiram Eldar, Dec 21 2024

Keywords

• nonn
• easy
• frac

Examples

			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, ...
		

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.

Crossrefs

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

Programs

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}]]]

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), ", "))};

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... .