A361365 - OEIS

%I #43 Dec 19 2024 11:45:36

%S 3,10,29,90,207,384,689,1226,2523,4446,7919

%N a(n) is the minimum possible sum of 2*n distinct positive numbers in a set, arranged in two subsets of size n each, such that the sum of any one element in each of the two subsets is a prime number.

%C Imagine a safe combination lock that has two independently rotating concentric number dials. The simplest model has just n = 1 number on each of its dials and one alignment position for those two numbers on them, allowing a single possible setting. But on the more advanced models there are n = 2, n = 3, and so on equally spaced numbers for each dial. All the numbers are different on a model, and when its dials are set in any combination, then every two numbers aligned will add up to a prime number. Necessarily, the numbers on one dial are all odd and on the other all even.

%C More than one set may correspond to some a(n), as is the case of a(4).

%C Any set corresponding to the n-th term is a partition of the term with 2*n distinct parts.

%C The procedure described in A162662 may aid the construction of candidate sets here, though their sums seem much greater than the minimum.

%H Kevin Ryde, <a href="/A361365/a361365.c.txt">C Code</a>

%e a(1) = 3, the minimum possible sum of two distinct numbers in the set. These add up to 3, which is a prime.

%e          1          2

%e a(2) = 10, the least possible sum of four distinct numbers in the set. Any number from the first column added to any number in the second, gives a prime number. There are 2^2 = 4 possibilities, e.g., 1 + 4 = 5, or 3 + 4 = 7, and so on.

%e          1          2

%e          3          4

%e a(3) = 29, the minimum possible sum of six distinct numbers in the set. Any number in the first column added to any number in the second, results in a prime number. There are 3^2 = 9 possibilities, e.g., 1 + 10 = 11, or 9 + 4 = 13, and so on.

%e          1          2

%e          3          4

%e          9         10

%e a(4) = 90 is attained in two different ways,

%e     sets     {1,5,11,17}  {2,6,12,36}

%e     or sets  {1,5,11,35}  {2,6,12,18}

%o (C) /* See links */

%Y Cf. A000040, A005843, A000290 (number of combinations).

%K nonn,more

%O 1,1

%A _Tamas Sandor Nagy_, Mar 09 2023

%E a(4)-a(11) from _Kevin Ryde_, Mar 23 2023