A340869 -id:A340869 - cp's OEIS frontend

A117330 a(n) is the determinant of the 3 X 3 matrix with entries the 9 consecutive primes starting with the n-th prime.

-78, 20, -36, 36, -40, -96, 96, -480, -424, 520, 348, 100, -540, 144, -144, -712, 240, 96, 480, -1120, -468, -1152, -3384, 1404, -576, -3924, 7884, -1548, -7312, 6288, -1828, -528, -768, 1920, 720, 768, -1920, 2400, -944, -9340, 12588, 15540, -864, 5600, 4124, -13668, -1428, 1552

Offset: 1

Cino Hilliard and Walter Kehowski, Apr 24 2006

The first term -78 is 6 mod 12 but all subsequent terms are 0,4,8 mod 12. Checked out to n=10000. A117329 is the subsequence formed by taking every 9th term.

The smallest absolute value of the sequence is 0.

			a(3)=-36 = det([[5,7,11],[13,17,19],[23,29,31]]).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A117329, A117345, A118799, A118815, A340869.

primedet := proc(n) local L; L:=map(ithprime,[$n..n+8]); linalg[det]([L[1..3],L[4..6],L[7..9]]) end;

Table[Det[Partition[Prime[Range[n,n+8]],3,3]],{n,50}] (* Harvey P. Dale, May 16 2019 *)

a(n) = matdet(matrix(3,3,i,j,prime((n+j-1)+3*(i-1)))); \\ Michel Marcus, Jan 25 2021

a(A117345(n)) = 0. - Hugo Pfoertner, Jan 26 2021

Edited by N. J. A. Sloane at the suggestion of Stefan Steinerberger, Jul 14 2007

A340874 Square root of the determinant of the 3 X 3 matrix [prime(k), prime(k+1), prime(k+2); prime(k+3), prime(k+4), prime(k+5); prime(k+6), prime(k+7), prime(k+8)] when that determinant is a square.

Original entry on oeis.org

6, 10, 12, 36, 294, 24, 0, 12, 24, 72, 0, 24, 12, 36, 0, 1564, 0, 12, 12, 0, 156, 0, 12, 60, 36, 48, 24, 0, 0, 72, 60, 60, 24, 60, 12, 0, 12, 12, 12, 0, 0, 12, 180, 0, 60, 0, 60, 72, 120, 0, 120, 0, 2150, 0, 24, 12, 0, 0, 60, 0, 36, 48, 120, 0, 0, 0, 0, 0, 0, 24, 0, 0, 56, 0, 24, 0, 48, 0, 2266

Offset: 1

J. M. Bergot and Robert Israel, Jan 24 2021

nonn

The prime k-tuples conjecture implies that, for example, there are infinitely many k for which the matrix is of the form [x, x+4, x+10; x+22, x+24, x+30; x+34, x+36, x+42], in which case the determinant is 12^2.

			a(3) = 12 because A340869(3) = 14 and the determinant of the 3 X 3 matrix [43, 47, 53; 59, 61, 67; 71, 73, 79] composed of prime(14) to prime(22) in order (by rows or columns) is 144 = 12^2.

Robert Israel, Table of n, a(n) for n = 1..3000

Cf. A117330, A340869.

f:= proc(n) local i,t;
t:= LinearAlgebra:-Determinant(Matrix(3, 3, [seq(ithprime(i), i=n..n+8)]));
if issqr(t) then sqrt(t) fi
end proc:
map(f, [$1..10000]);

m = 10^4; p = Prime[Range[m + 8]]; Select[Table[Sqrt @ Det @ Partition[p[[n ;; n + 8]], 3], {n, 1, m}], IntegerQ] (* Amiram Eldar, Jan 25 2021 *)

f(n) = matdet(matrix(3,3,i,j,prime((n+j-1)+3*(i-1)))); \\ A117330
lista(nn) = my(x); for (n=1, nn, if (issquare(f(n), &x), print1(x, ", "))); \\ Michel Marcus, Jan 25 2021

a(n) = sqrt(A117330(A340869(n))).

cp's OEIS Frontend

A117330 a(n) is the determinant of the 3 X 3 matrix with entries the 9 consecutive primes starting with the n-th prime.

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