A240986 Determinants of n X n matrices of sets of distinct primes selected by increasing prime gaps (see comments).

3, 6, -36, -216, 1296, -5184, -145152, -3856896, -170325504, -6133211136, 1094593056768, 26742290558976, -497681937801216, -14357497419546624, 657148066947072000, 12008320398059765760, 1322255096225695531008, 70546799432003423698944, -6537119853797882157072384, -27940593871362459110473728

Offset: 1

Author

Samuel J. Erickson, Aug 06 2014

Keywords

• sign

Comments

Let P = {3,5,7,11,...} be the sequence of odd primes and let P(k) = {prime in P: (prime+2k) is in P} (although set builder notation is used for P(k) we will still assume that P(k) is a sequence). Let M(n) be the n X n matrix where row 1 is the first n elements from P(1), row 2 is the first n elements from P(2), and in general row j is the first n elements from P(j). This sequence is the sequence of determinants for M(1), M(2), M(3), M(4), ..., M(9).

Examples

			For the first element of the sequence we find the determinant of the matrix [[3,5],[3,7]], where [3,5] is row 1 and [3,7] is row 2. These numbers are there because in row 1 we are looking at the primes where we can add 2 to get another prime; 3+2 is prime and so is 5+2, so they go in row 1. Similarly, for the second row we get [3,7] because these are the first primes such that when 4 is added we get a prime: 3+4 and 7+4 are both prime, so they go in row 2. For the second entry in the sequence we take the determinant of [[3,5,11],[3,7,13],[5,7,11]]; the reason we get [5,7,11] in the third row is because 5 is the first prime where 5+6 is prime, 7 is second prime where 7+6 is prime, and 11 is the third prime where 11+6 is prime.
		

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..200
• Samuel J. Erickson, Python Code
• Sachin Joglekar, Determinant of matrix of any order (Python)

Crossrefs

Cf. A001359, A023200, A023201, A023202, A023203.

Programs

PARI

a(n) = {my(m=matrix(n,n), j); for (i=1, n, j = 1; forprime(p=2, , if (isprime(p+2*i), m[i,j] = p; j++); if (j > n, break););); matdet(m);} \\ Michel Marcus, May 04 2019

Python

# See Erickson link.

Extensions

Offset 1 and more terms from Michel Marcus, May 04 2019