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URL: https://oeis.org/A339105

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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.
1
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
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
KEYWORD
nonn
AUTHOR
J. M. Bergot and Robert Israel, Nov 23 2020
STATUS
approved