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A360264 Sum of mass(k/n) for all k, 1 <= k <= n, that are relatively prime to n.

%I A360264 #26 Dec 13 2024 10:25:21
%S A360264 1,2,6,8,18,12,34,26,42,32,74,36,98,56,80,78,150,64,178,92,140,116,
%T A360264 238,100,238,148,222,160,338,112,374,214,280,220,348,180,486,260,356,
%U A360264 248,562,192,602,316,388,344,682,264,662,328,528,404,810,308,688,424
%N A360264 Sum of mass(k/n) for all k, 1 <= k <= n, that are relatively prime to n.
%C A360264 The mass of a rational k/n is the sum of the partial quotients in the continued fraction for k/n.  Alternatively, it is the number of steps in the "subtractive algorithm" to compute gcd(k,n).
%H A360264 Amiram Eldar, <a href="/A360264/b360264.txt">Table of n, a(n) for n = 1..10000</a>
%H A360264 Christoph Aistleitner, Bence Borda, and Manuel Hauke, <a href="https://arxiv.org/abs/2210.14095">On the distribution of partial quotients of reduced fractions with fixed denominator</a>, ArXiv preprint arXiv:2210.14095 [math.NT], 2022-2023.
%H A360264 Bernhard Liehl, <a href="https://doi.org/10.1007/BF01192762">Über die Teilnenner endlicher Kettenbrüche</a>, Arch. Math. (Basel), 40 (1983), 139-147.
%H A360264 A. A. Panov, <a href="https://www.mathnet.ru/php/archive.phtml?wshow=paper&amp;jrnid=mzm&amp;paperid=5989&amp;option_lang=eng">The mean for a sum of elements in a class of finite continued fractions</a> (in Russian), Mat. Zametki 32 (1982), 593-600, 747; <a href="https://doi.org/10.1007/BF01358470">English version</a>, Mathematical Notes of the Academy of Sciences of the USSR 32 (1982), 781-785.
%H A360264 Maurice Shrader-Frechette, <a href="https://www.jstor.org/stable/2690435">Modified Farey sequences and continued fractions</a>, Math. Mag., 54 (1981), 60-63.
%F A360264 Panov (1982) and Liehl (1983) independently proved that a(n) is asymptotically (6/Pi)^2*n*(log n)^2.
%e A360264 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.
%p A360264 a:= n-> add(`if`(igcd(n, k)=1, add(i, i=convert(k/n, confrac)), 0), k=1..n):
%p A360264 seq(a(n), n=1..60);  # _Alois P. Heinz_, Jan 31 2023
%t A360264 a[n_] := Total@ Flatten@ (ContinuedFraction[#/n] & /@ Select[Range[n], CoprimeQ[#, n] &]); Array[a, 100] (* _Amiram Eldar_, Dec 13 2024 *)
%Y A360264 Cf. A000010, A049835.
%K A360264 nonn
%O A360264 1,2
%A A360264 _Jeffrey Shallit_, Jan 31 2023