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A383621 a(n) is the minimum possible value of x_1 + x_2 + ... + x_n where x_1, x_2, ..., x_n are positive integers such that x_i does not divide x_j for any i != j.

%I A383621 #25 Jul 01 2025 19:04:26
%S A383621 1,5,10,17,28,41,55,72,91,111,134,159,187,216,247,282,319,360,403,447,
%T A383621 493,540,589,641,694,749,808,869,934,1001,1069,1139,1210,1283,1359,
%U A383621 1436,1515,1598,1683,1772,1863,1955,2050,2147,2245,2345,2446,2549,2656,2765,2878
%N A383621 a(n) is the minimum possible value of x_1 + x_2 + ... + x_n where x_1, x_2, ..., x_n are positive integers such that x_i does not divide x_j for any i != j.
%C A383621 The sequence is the solution to Problem 9 of 2022 Chinese Team Selection Test. - _Yifan Xie_, Jun 27 2025
%H A383621 Yifan Xie, <a href="/A383621/b383621.txt">Table of n, a(n) for n = 1..10000</a>
%H A383621 Yifan Xie, <a href="/A383621/a383621_1.pdf">On a solvable minimal sum antichain problem</a>
%F A383621 a(n) ~ c*n^2, where c = (3/2)*(Sum_{k=0..oo} 1/A027649(k))^(-1) = 1.108410...
%e A383621 For n <= 6, the construction is given by the n smallest primes.
%e A383621 For n = 7, the numbers 4, 5, 6, 7, 9, 11, 13 are mutually indivisible and their sum is a(7) = 55.
%o A383621 (PARI) A027649(n) = 2*3^n-2^n;
%o A383621 A383622(nn) = {my(v=[]); for(n=0, logint(nn,3), d = A027649(n); m = floor(nn/d); for(i=0, floor(m/6), if(6*i+1 <= m, v=concat(v, d*(6*i+1))); if(6*i+5 <= m, v=concat(v, d*(6*i+5))))); v=vecsort(v); v};
%o A383621 lista(nn) = {u = A383622(3*nn); my(v=vector(nn)); s=0; for(n=1, nn, s = s + u[n]; v[n] = s); v};
%Y A383621 Cf. A027649.
%Y A383621 Partial sums of A383622.
%K A383621 nonn,easy
%O A383621 1,2
%A A383621 _Yifan Xie_, May 10 2025