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A240986
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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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1
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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
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1,1
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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.
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PROG
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(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
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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