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%I #13 Nov 24 2020 16:38:18
%S 746640,384984,291960,209304,163224,133272,111384,102240,89712,77328,
%T 69480,59400,59904,55080,48456,44712,43704,40104,38088,37440,35856,
%U 35208,31968,31968,29736,30816,27216,25488,22968,23544,21672,20808,19872,19584,19296,20808,19368,17640,17208,17136,16128
%N 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.
%C By symmetry (permutation of rows and of columns, and transposition) all terms are divisible by 72.
%C Presumably most entries are 0 when n is large. The first occurrence of 0 is a(1229).
%H Robert Israel, <a href="/A339105/b339105.txt">Table of n, a(n) for n = 1..10000</a>
%e For n=1, one of the matrices is
%e [ 2 3 5 ]
%e [ 7 11 13 ]
%e [ 17 19 23 ]
%e 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).
%p f:= proc(n) local M,P,p;
%p p:= [seq(ithprime(n+i),i=0..8)];
%p 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])));
%p M:= map(t -> LinearAlgebra:-Determinant(Matrix(3,3,t)), P);
%p 72*numboccur(0, [seq(seq(M[i] mod p[j],i=1..5040),j=1..9)])
%p end proc:
%p map(f, [$1..50]);
%K nonn
%O 1,1
%A _J. M. Bergot_ and _Robert Israel_, Nov 23 2020
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