cp's OEIS Frontend

A180127 Upper bound for the determinant of an n X n matrix whose elements are a permutation of the first n^2 prime numbers.

%I A180127 #8 Aug 28 2021 06:21:19
%S A180127 2,32,7414,4993844,5761178228,11320943775475,35966786849223443,
%T A180127 154715716383037989022,1041732064414822689366009,
%U A180127 8436103376958505162325231670,95816938885687281564299004113250,1337411611273240103793149357629547975,24089834168067078066162508828810807131186
%N A180127 Upper bound for the determinant of an n X n matrix whose elements are a permutation of the first n^2 prime numbers.
%C A180127 a(n) is an upper bound for A180128(n).
%H A180127 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 A180127 Markus Sigg, <a href="https://arxiv.org/abs/1804.02897">Gasper's determinant theorem, revisited</a>, arXiv:1804.02897 [math.CO], 2018.
%F A180127 Let c = A007504(n^2)/n [(1/n)*sum of first n^2 primes]
%F A180127 and d = A024450(n^2)/n [(1/n)*sum of first n^2 squares of primes]
%F A180127 Then a(n) = floor(c*sqrt((d-t)^(n-1))) with t = (c^2-d)/(n-1).
%F A180127 log(a(n)) ~ (5*log(n) - log(3))*n/2 + n*log(log(n)). - _Vaclav Kotesovec_, Aug 28 2021
%o A180127 (PARI) a180127(n)={if(n<2,2, my(c=sum(k=1,n^2,prime(k))/n, d=sum(k=1,n^2,prime(k)^2)/n, t=(c^2-d)/(n-1)); floor(c*sqrt((d-t)^(n-1))))} \\ _Hugo Pfoertner_, Aug 27 2021
%Y A180127 Cf. A180128 [Maximal determinant of matrix with first n^2 primes], A085000 [Maximal determinant of matrix with elements 1, ..., n^2], A180087 [Upper bound for A085000], A007504 [Sum of first n primes], A024450 [Sum of first n squares of primes].
%K A180127 nonn
%O A180127 1,1
%A A180127 _Hugo Pfoertner_, Aug 12 2010