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.

1, 5, 10, 17, 28, 41, 55, 72, 91, 111, 134, 159, 187, 216, 247, 282, 319, 360, 403, 447, 493, 540, 589, 641, 694, 749, 808, 869, 934, 1001, 1069, 1139, 1210, 1283, 1359, 1436, 1515, 1598, 1683, 1772, 1863, 1955, 2050, 2147, 2245, 2345, 2446, 2549, 2656, 2765, 2878

Offset: 1

Author

Yifan Xie, May 10 2025

Keywords

• nonn
• easy

Comments

The sequence is the solution to Problem 9 of 2022 Chinese Team Selection Test. - Yifan Xie, Jun 27 2025

Examples

			For n <= 6, the construction is given by the n smallest primes.
For n = 7, the numbers 4, 5, 6, 7, 9, 11, 13 are mutually indivisible and their sum is a(7) = 55.
		

Links

• Yifan Xie, Table of n, a(n) for n = 1..10000
• Yifan Xie, On a solvable minimal sum antichain problem

Crossrefs

Cf. A027649.

Partial sums of A383622.

Programs

PARI

A027649(n) = 2*3^n-2^n;
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};
lista(nn) = {u = A383622(3*nn); my(v=vector(nn)); s=0; for(n=1, nn, s = s + u[n]; v[n] = s); v};

Formula

a(n) ~ c*n^2, where c = (3/2)*(Sum_{k=0..oo} 1/A027649(k))^(-1) = 1.108410...