

A240986


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


1



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

1,1


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).


LINKS

Table of n, a(n) for n=1..20.
Samuel J. Erickson, Python Code
Sachin Joglekar, Determinant of matrix of any order (Python)


EXAMPLE

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.


PROG

(Python) See link for code.
(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


CROSSREFS

Cf. A001359, A023200, A023201, A023202, A023203.
Sequence in context: A068084 A003674 A211895 * A120595 A048642 A264702
Adjacent sequences: A240983 A240984 A240985 * A240987 A240988 A240989


KEYWORD

sign,more


AUTHOR

Samuel J. Erickson, Aug 06 2014


EXTENSIONS

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


STATUS

approved



