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A256391 a(n) = number of tuples (a,b,c,d) of natural numbers a,b,c,d <= n with gcd(a,b)=gcd(b,c)=gcd(c,d)=gcd(d,a)=1.

%I A256391 #27 Jun 09 2018 10:00:00
%S A256391 1,7,35,79,243,319,787,1155,1859,2295,4267,4891,8295,9743,11851,14539,
%T A256391 22191,24359,35427,39387,45915,51687,71171,76407,94911,105047,123251,
%U A256391 134447,174003,180835,229783,253007,281447,305111,343315,360215,442547,476115,523111,552307
%N A256391 a(n) = number of tuples (a,b,c,d) of natural numbers a,b,c,d <= n with gcd(a,b)=gcd(b,c)=gcd(c,d)=gcd(d,a)=1.
%C A256391 The sequence has the asymptotics a(n) = rho*n^4 + O(n^3*log^2(n)) where rho=prod_p(1 - 4/p^2 + 4/p^3 - 1/p^4) = 0.21777871661953... (product extended to primes). See A256392.
%H A256391 Juan Arias-de-Reyna, <a href="/A256391/b256391.txt">Table of n, a(n) for n = 1..100</a>
%H A256391 J. Arias de Reyna and R. Heyman, <a href="http://arxiv.org/abs/1403.2769">Counting tuples restricted by pairwise primality</a>, arXiv:1403.2769 [math.NT], 2014.
%H A256391 J. Arias de Reyna, R. Heyman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Heyman/heyman6.html">Counting Tuples Restricted by Pairwise Coprimality Conditions</a>, J. Int. Seq. 18 (2015) 15.10.4
%F A256391 a(n) = sum_a sum_b sum_c sum_d mu(a) mu(b) mu(c) mu(d) [n/gcd(a,b)][n/gcd(b,c)][n/gcd(c,d)][n/gcd(d,a)], where mu is Moebius function, a,b,c,d run through natural numbers.
%e A256391 For n=2, a(2)=7 counting the tuples (1,1,1,1), (2,1,1,1), (1,2,1,1), (1,1,2,1), (1,1,1,2), (2,1,2,1), (1,2,1,2).
%t A256391 A[M_] := A[M] = Module[{X, a1, a2, a3, a4, K, count, k},
%t A256391     X = Flatten[
%t A256391       Table[{a1, a2, a3, a4}, {a1, 1, M}, {a2, 1, M}, {a3, 1, M}, {a4,
%t A256391          1, M}], 3];
%t A256391     K = Length[X];
%t A256391     count = 0;
%t A256391     For[k = 1, k <= K, k++,
%t A256391      {a1, a2, a3, a4} = X[[k]];
%t A256391      If[(GCD[a1, a2] == 1) && (GCD[a2, a3] == 1) && (GCD[a3, a4] ==
%t A256391          1) && (GCD[a4, a1] == 1), count = count + 1]];
%t A256391     count];
%t A256391 Table[A[n], {n, 1, 20}]
%Y A256391 Cf. A256390.
%K A256391 nonn
%O A256391 1,2
%A A256391 _Juan Arias-de-Reyna_, Mar 27 2015