A383621 - OEIS

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

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OFFSET

1,2

COMMENTS

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

LINKS

Yifan Xie, Table of n, a(n) for n = 1..10000

Yifan Xie, On a solvable minimal sum antichain problem

FORMULA

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

EXAMPLE

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.

PROG

(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};

CROSSREFS

Cf. A027649.

Partial sums of A383622.

Sequence in context: A306011 A080341 A271328 * A086653 A215449 A174062

Adjacent sequences: A383618 A383619 A383620 * A383622 A383623 A383624

KEYWORD

nonn,easy

AUTHOR

Yifan Xie, May 10 2025

STATUS

approved