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

1,1

Comments

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

Links

Table of n, a(n) for n=1..13.

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.

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

Prog

(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

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].

Sequence in context: A247859 A202629 A129349 * A342099 A091804 A012853

Adjacent sequences: A180124 A180125 A180126 * A180128 A180129 A180130

Keyword

nonn

Author

Hugo Pfoertner, Aug 12 2010

Status

approved