A317679 -id:A317679 - cp's OEIS frontend

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 = np_j - Sum_{k=1..n,k!=j} p_k, j=1..n such that the set union {p_j} U {q_j} contains 2n 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

Offset: 1

Hugo Pfoertner, Aug 10 2018

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.

			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
  6   660    97 101 103 107 113 139   19  47  61  89 131 313

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

Cf. A024939, A317679, A317680.

A317680 Least sum S_n of n distinct primes p_i such that q_j = np_j - Sum_{k=1..n,k!=j} p_k, j=1..n is prime, such that the set union {p_i} U {q_i} contains 2n distinct primes.

Original entry on oeis.org

28, 141, 138, 395, 660, 1425, 1960, 2217, 3810, 5005, 7230

Offset: 2

Hugo Pfoertner, Aug 10 2018

The sets of primes p_i and q_i and a description of the problem are provided in A317678 and A317679. a(1) is omitted to indicate the nonexistence of a solution.

a(13) <= 8955, a(14) <= 12122.

Cf. A317678, A317679.

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A317680 Least sum S_n of n distinct primes p_i such that q_j = np_j - Sum_{k=1..n,k!=j} p_k, j=1..n is prime, such that the set union {p_i} U {q_i} contains 2n distinct primes.

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A317680 Least sum S_n of n distinct primes p_i such that q_j = n*p_j - Sum_{k=1..n,k!=j} p_k, j=1..n is prime, such that the set union {p_i} U {q_i} contains 2*n distinct primes.

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A317680 Least sum S_n of n distinct primes p_i such that q_j = np_j - Sum_{k=1..n,k!=j} p_k, j=1..n is prime, such that the set union {p_i} U {q_i} contains 2n distinct primes.