A360264 Sum of mass(k/n) for all k, 1 <= k <= n, that are relatively prime to n.
1, 2, 6, 8, 18, 12, 34, 26, 42, 32, 74, 36, 98, 56, 80, 78, 150, 64, 178, 92, 140, 116, 238, 100, 238, 148, 222, 160, 338, 112, 374, 214, 280, 220, 348, 180, 486, 260, 356, 248, 562, 192, 602, 316, 388, 344, 682, 264, 662, 328, 528, 404, 810, 308, 688, 424
Offset: 1
Keywords
Examples
For n = 4 the two numbers relatively prime to n are 1 and 3; 1/4 = [0,4] and 3/4 = [0,1,3]. So the sum of all these is a(3) = 8.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
- Christoph Aistleitner, Bence Borda, and Manuel Hauke, On the distribution of partial quotients of reduced fractions with fixed denominator, ArXiv preprint arXiv:2210.14095 [math.NT], 2022-2023.
- Bernhard Liehl, Über die Teilnenner endlicher Kettenbrüche, Arch. Math. (Basel), 40 (1983), 139-147.
- A. A. Panov, The mean for a sum of elements in a class of finite continued fractions (in Russian), Mat. Zametki 32 (1982), 593-600, 747; English version, Mathematical Notes of the Academy of Sciences of the USSR 32 (1982), 781-785.
- Maurice Shrader-Frechette, Modified Farey sequences and continued fractions, Math. Mag., 54 (1981), 60-63.
Programs
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Maple
a:= n-> add(`if`(igcd(n, k)=1, add(i, i=convert(k/n, confrac)), 0), k=1..n): seq(a(n), n=1..60); # Alois P. Heinz, Jan 31 2023
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Mathematica
a[n_] := Total@ Flatten@ (ContinuedFraction[#/n] & /@ Select[Range[n], CoprimeQ[#, n] &]); Array[a, 100] (* Amiram Eldar, Dec 13 2024 *)
Formula
Panov (1982) and Liehl (1983) independently proved that a(n) is asymptotically (6/Pi)^2*n*(log n)^2.
Comments