A361365 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.

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

Offset: 1

Author

Tamas Sandor Nagy, Mar 09 2023

Keywords

• nonn
• more

Comments

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.

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

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

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

Examples

			a(1) = 3, the minimum possible sum of two distinct numbers in the set. These add up to 3, which is a prime.
         1          2
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.
         1          2
         3          4
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.
         1          2
         3          4
         9         10
a(4) = 90 is attained in two different ways,
    sets     {1,5,11,17}  {2,6,12,36}
    or sets  {1,5,11,35}  {2,6,12,18}
		

Links

• Kevin Ryde, C Code

Crossrefs

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

Programs

C

/* See links */

Extensions

a(4)-a(11) from Kevin Ryde, Mar 23 2023