A339105 - OEIS

%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