A339105 Consider all 3 X 3 matrices M whose entries are the n-th to (n+8)-th primes prime(n), ..., prime(n+8), in any order. a(n) is the sum of the number of M such that det(M) is divisible by prime(n+i), for i from 0 to 8.

746640, 384984, 291960, 209304, 163224, 133272, 111384, 102240, 89712, 77328, 69480, 59400, 59904, 55080, 48456, 44712, 43704, 40104, 38088, 37440, 35856, 35208, 31968, 31968, 29736, 30816, 27216, 25488, 22968, 23544, 21672, 20808, 19872, 19584, 19296, 20808, 19368, 17640, 17208, 17136, 16128

Offset

1,1

Comments

By symmetry (permutation of rows and of columns, and transposition) all terms are divisible by 72.

Presumably most entries are 0 when n is large. The first occurrence of 0 is a(1229).

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

For n=1, one of the matrices is
[  2  3  5 ]
[  7 11 13 ]
[ 17 19 23 ]
which has determinant -78.  This is divisible by three of the primes forming the matrix's entries, namely 2, 3 and 13, so it contributes 3 to a(n).

Maple

f:= proc(n) local M, P, p;
  p:= [seq(ithprime(n+i), i=0..8)];
  P:= map(t -> [p[1], op(t)], select(t -> t[1] < t[2] and t[3] < t[6] and t[1] < t[3], combinat:-permute(p[2..9])));
  M:= map(t -> LinearAlgebra:-Determinant(Matrix(3, 3, t)), P);
  72*numboccur(0, [seq(seq(M[i] mod p[j], i=1..5040), j=1..9)])
end proc:
map(f, [$1..50]);

Crossrefs

Sequence in context: A133183 A119403 A184149 * A133979 A352345 A205743

Adjacent sequences: A339102 A339103 A339104 * A339106 A339107 A339108

Keyword

nonn

Author

J. M. Bergot and Robert Israel, Nov 23 2020

Status

approved