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Determinants of n X n matrices of sets of distinct primes selected by increasing prime gaps (see comments).
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%I #48 Mar 02 2020 00:43:33

%S 3,6,-36,-216,1296,-5184,-145152,-3856896,-170325504,-6133211136,

%T 1094593056768,26742290558976,-497681937801216,-14357497419546624,

%U 657148066947072000,12008320398059765760,1322255096225695531008,70546799432003423698944,-6537119853797882157072384,-27940593871362459110473728

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

%C 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).

%H Jinyuan Wang, <a href="/A240986/b240986.txt">Table of n, a(n) for n = 1..200</a>

%H Samuel J. Erickson, <a href="/A240986/a240986.txt">Python Code</a>

%H Sachin Joglekar, <a href="http://code.activestate.com/recipes/578108-determinant-of-matrix-of-any-order">Determinant of matrix of any order (Python)</a>

%e 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.

%o (Python) See link for code.

%o (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

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

%K sign

%O 1,1

%A _Samuel J. Erickson_, Aug 06 2014

%E Offset 1 and more terms from _Michel Marcus_, May 04 2019