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%I #18 Jan 19 2022 19:27:01
%S 1,2,29,6640,4868296,5725998504,11305600374272,35954639671827328
%N Maximal determinant of an n X n matrix whose elements are a permutation of the first n^2 prime numbers.
%C The terms a(5), a(6), a(7) were found by tabu search, with strong numerical evidence for the optimality of a(7).
%C A known lower bound for the next term a(8) is 154665569137423060000.
%C Upper bounds for higher terms can be found by the method described by O. Gasper, H. Pfoertner and M. Sigg, and are given in A180127, e.g., a(8) <= 154715716383037989022.
%C An improved lower bound is a(8) >= 154671943501236284416, provided in a private communication by Richard Gosiorovsky. - _Hugo Pfoertner_, Aug 27 2021
%H Ortwin Gasper, Hugo Pfoertner and Markus Sigg, <a href="http://www.emis.de/journals/JIPAM/article1119.html?sid=1119">An Upper Bound for the Determinant of a Matrix with given Entry Sum and Square Sum</a> JIPAM, Vol. 10, Iss. 3, Art. 63, 2008
%H Markus Sigg, <a href="https://arxiv.org/abs/1804.02897">Gasper's determinant theorem, revisited</a>, arXiv:1804.02897 [math.CO]
%H <a href="/index/De#determinants">Index entries for sequences related to maximal determinants</a>
%e a(2) = 29:
%e . 7 3
%e . 2 5
%e a(3) = 6640:
%e . 23 11 5
%e . 3 17 13
%e . 7 2 19
%e a(4) = 4868296:
%e . 53 11 23 13
%e . 17 47 29 3
%e . 7 5 43 37
%e . 19 31 2 41
%e a(5) = 5725998504
%e . 89 41 23 2 53
%e . 31 97 29 47 11
%e . 59 13 79 61 7
%e . 37 19 5 83 67
%e . 3 43 71 17 73
%e a(6) = 11305600374272:
%e . 137 73 7 89 83 13
%e . 79 139 67 19 3 97
%e . 101 5 149 61 37 53
%e . 2 109 103 71 113 11
%e . 59 29 41 17 131 127
%e . 23 47 43 151 31 107
%e a(7) = 35954639671827332:
%e . 227 71 173 43 83 29 73
%e . 151 163 5 181 2 103 89
%e . 31 223 139 61 137 97 13
%e . 23 47 157 211 109 19 131
%e . 113 7 67 127 167 199 17
%e . 53 79 149 37 11 193 179
%e . 101 107 3 41 191 59 197
%Y Cf. A180127 [upper bounds for a(n)], A085000 [maximal determinants for matrix elements 1, ..., n^2].
%Y Cf. A340923, A340924, A340925.
%K nonn,hard,more
%O 0,2
%A _Hugo Pfoertner_, Aug 11 2010
%E a(7) corrected, based on private communication from Richard Gosiorovsky by _Hugo Pfoertner_, Aug 27 2021
%E a(0)=1 prepended by _Alois P. Heinz_, Jan 19 2022
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