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

2, 32, 7414, 4993844, 5761178228, 11320943775475, 35966786849223443, 154715716383037989022, 1041732064414822689366009, 8436103376958505162325231670, 95816938885687281564299004113250, 1337411611273240103793149357629547975, 24089834168067078066162508828810807131186

Offset: 1

Author

Hugo Pfoertner, Aug 12 2010

Keywords

• nonn

Comments

a(n) is an upper bound for A180128(n).

Links

• Ortwin Gasper, Hugo Pfoertner and Markus Sigg, An Upper Bound for the Determinant of a Matrix with given Entry Sum and Square Sum JIPAM, Vol. 10, Iss. 3, Art. 63, 2008
• Markus Sigg, Gasper's determinant theorem, revisited, arXiv:1804.02897 [math.CO], 2018.

Crossrefs

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].

Programs

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

Formula

Let c = A007504(n^2)/n [(1/n)*sum of first n^2 primes]

and d = A024450(n^2)/n [(1/n)*sum of first n^2 squares of primes]

Then a(n) = floor(c*sqrt((d-t)^(n-1))) with t = (c^2-d)/(n-1).

log(a(n)) ~ (5*log(n) - log(3))*n/2 + n*log(log(n)). - Vaclav Kotesovec, Aug 28 2021