A088922 Consider the n X n matrix with entries (i*j mod n), where i,j=0..n-1; a(n) = rank of this matrix over the real numbers.

0, 1, 2, 3, 3, 5, 4, 6, 6, 7, 6, 10, 7, 9, 10, 11, 9, 13, 10, 14, 13, 13, 12, 18, 14, 15, 16, 18, 15, 21, 16, 20, 19, 19, 20, 25, 19, 21, 22, 26, 21, 27, 22, 26, 27, 25, 24, 32, 26, 29, 28, 30, 27, 33, 30, 34, 31, 31, 30, 40, 31, 33, 36, 37, 35, 39, 34, 38, 37, 41, 36, 46, 37, 39, 42, 42, 41, 45, 40, 48, 44, 43, 42, 52, 45, 45, 46, 50, 45, 55, 48, 50, 49, 49, 50, 58, 49, 53, 54, 57

Offset

1,3

Comments

Possibly related to Maillet's determinants.

Links

Table of n, a(n) for n=1..100.

Alexander Adam, Proof of the formula

Wikipedia, Maillet's determinant

Formula

Let n = Prod_{i>0} p_i^{m_i} be the prime factorization of n. Then a(n) = floor((n + 1)/2) + Prod_{i>0} (m_i + 1) - 2. - Alexander Adam, Nov 10 2012

a(n) = A000005(n) + A110654(n) - 2.

Example

From Alexander Adam, Nov 10 2012, (Start)
a(2^m) = 2^(m-1) + m - 1.
Let p >= 3 be a prime number. Then a(p^m) = (p^m + 1) / 2 + m - 1.
a(625000) = a(2^3*5^7) = 2^2*5^7 + 4 * 8 - 2 = 312530. (end)

Mathematica

a[n_] := MatrixRank[Table[Table[Mod[i * j, n], {j, 0, n - 1}], {i, 0, n - 1}]]; Array[a, 100] (* Alexander Adam, Nov 10 2012 *)

Prog

(PARI) a(n) = matrank(matrix(n, n, i, j, (i*j)%n))

Crossrefs

Cf. A218322, A055513, A203411, A000927.

Sequence in context: A088241 A163126 A304818 * A357658 A143092 A143089

Adjacent sequences: A088919 A088920 A088921 * A088923 A088924 A088925

Keyword

nonn

Author

Max Alekseyev, Dec 01 2003

Status

approved