A317678 - OEIS

A317678

Sets of n distinct primes p_i in ascending order, written as triangle, such that S_n = Sum_{k=1..n} p_k is minimized and that there exists a compensation set of n primes q_j with q_j = n*p_j - Sum_{k=1..n,k!=j} p_k, j=1..n such that the set union {p_j} U {q_j} contains 2*n distinct primes and Sum_{k=1..n} q_k = S_n.

1, 11, 17, 37, 43, 61, 29, 31, 37, 41, 67, 71, 73, 83, 101, 97, 101, 103, 107, 113, 139, 179, 181, 191, 199, 211, 223, 241, 223, 227, 229, 233, 239, 257, 269, 283, 223, 227, 229, 233, 239, 241, 251, 257, 317, 347, 349, 353, 359, 367, 373, 397, 401, 421, 443

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OFFSET

1,2

COMMENTS

In case of ties, i.e., if more than one set exists for the same minimal sum S_n, the lexicographically least set is chosen. Since the condition of distinctness of {p_j} U {q_j} cannot be satisfied for n=1, the sets p = q = {1} are assumed to complete the triangle.

The minimal sums S_n are provided in A317680 and the corresponding compensation sets are provided in A317679. The compensation set is a solution to the "fair compensation" problem. n persons who own individual shares of p_i units of an asset agree to equally share those assets among themselves and a person n+1, who owns 0 units of this asset, but is willing to pay S_n units of compensation, e.g. money, to buy his share of S_n/(n+1) units of the asset. S_n/(n+1) doesn't need to be integer. q_j as defined above is a fair compensation for person j's relinquishment of assets by equipartitioning, assuming a constant price per asset.

LINKS

Table of n, a(n) for n=1..55.

IBM Research, Ponder This August 2018 - Challenge, 9th row of table is one of many solutions.

EXAMPLE

Table begins:

n Sum p_k q_k

A317680 A317679

1 1 1 1

2 28 11 17 5 23

3 141 37 43 61 7 31 103

4 138 29 31 37 41 7 17 47 67

5 395 67 71 73 83 101 7 31 43 103 211