cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

-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

Views

Author

Cino Hilliard and Walter Kehowski, Apr 24 2006

Keywords

Comments

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.

Examples

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

Links

Crossrefs

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

Programs

  • Maple
    primedet := proc(n) local L; L:=map(ithprime,[$n..n+8]); linalg[det]([L[1..3],L[4..6],L[7..9]]) end;
  • Mathematica
    Table[Det[Partition[Prime[Range[n,n+8]],3,3]],{n,50}] (* Harvey P. Dale, May 16 2019 *)
  • PARI
    a(n) = matdet(matrix(3,3,i,j,prime((n+j-1)+3*(i-1)))); \\ Michel Marcus, Jan 25 2021

Formula

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

Extensions

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

A340869 Numbers k such that 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)] is a square.

Original entry on oeis.org

4, 12, 14, 131, 222, 229, 330, 351, 356, 525, 561, 825, 969, 979, 1009, 1115, 1123, 1243, 1722, 1826, 2221, 2632, 2673, 2814, 3167, 3436, 3437, 3966, 4056, 4307, 4583, 5010, 5137, 5509, 5772, 6031, 6034, 6230, 6233, 6363, 6505, 6532, 6794, 7112, 7551, 8154, 8330, 8476, 9260, 9348, 9349, 9613
Offset: 1

Views

Author

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

Keywords

Comments

Numbers k such that A117330(k) is a square.

Examples

			a(3) = 14 is a term because A117330(14) = Determinant([43,47,53; 59,61,67; 71,73,79]) = 144 = 12^2.

Links

Crossrefs

Cf. A117330, A340874. Includes A117345.

Programs

  • Maple
    f:= proc(n) local i; LinearAlgebra:-Determinant(Matrix(3,3,[seq(ithprime(i),i=n..n+8)])) end proc:
select(t -> issqr(f(t)), [$1..10000]);
  • Mathematica
    okQ[k_] := IntegerQ@ Sqrt@ Det@ Partition[Prime[k+#]& /@ Range[0, 8], 3];
Select[Range[10000], okQ] (* Jean-François Alcover, Feb 10 2023 *)
  • PARI
    isok(k) = issquare(matdet(matrix(3,3,i,j,prime((k+j-1)+3*(i-1))))); \\ Michel Marcus, Jan 25 2021
  • Python
    from sympy import nextprime, Matrix, integer_nthroot
k,A340869_list, plist = 1,[], [2, 3, 5, 7, 11, 13, 17, 19, 23]
while k < 10**7:
    d = Matrix(plist).reshape(3,3).det()
    if d >= 0 and integer_nthroot(d,2)[1]:
        A340869_list.append(k)
    k,plist = k+1,plist[1:]+[nextprime(plist[-1])] # Chai Wah Wu, Jan 25 2021

A117359 Indices n == 1 (mod 9) such that the 3 X 3 matrix with components (row by row) prime(n+k), 0 <= k <= 8, has zero determinant.

Original entry on oeis.org

1009, 6031, 9613, 19378, 49996, 67285, 91549, 101278, 102097, 107182, 142723, 154792, 168562, 175006, 183718, 196345, 200530, 204031, 215407, 240292, 263395, 264628, 277723, 289171, 299323, 307684, 313111, 369676, 372601, 376921, 425935
Offset: 1

Views

Author

Cino Hilliard, Apr 24 2006

Keywords

Comments

By considering only indices congruent to 1 (mod 9) each prime occurs in exactly one of these matrices. - Subsequence of A117345.

Crossrefs

Cf. A117345.

Programs

  • PARI
    {m=426000;forstep(n=1,m,9,M=matrix(3,3,i,j,prime(n+3*(i-1)+j-1));if(matdet(M,1)==0,print1(n,",")))}

Extensions

Edited by Klaus Brockhaus, Apr 28 2006

A337160 Primes p such that the 3 X 3 matrix with components (row by row) prime(k+m), 0 <= m <= 8 has zero determinant, where p = prime(k).

Original entry on oeis.org

2213, 4073, 8011, 9041, 15649, 23663, 37483, 38453, 59663, 63487, 65111, 71861, 83557, 97157, 100279, 118801, 129527, 131707, 139291, 163601, 166597, 166799, 180181, 180233, 195691, 203807, 209233, 217201, 227561, 238657, 289139, 309121, 327473
Offset: 1

Views

Author

Jianing Song, Jan 28 2021

Keywords

Comments

Primes arising from A117345.

Examples

			The next 8 primes after 2213 are 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, and we have det({{2213, 2221, 2237}, {2239, 2243, 2251}, {2267, 2269, 2273}}) = 0, hence 2213 is a term.

Crossrefs

Cf. A117330, A117345, A340923.

Programs

  • PARI
    for(k=1, 35000, M=matrix(3, 3, i, j, prime(k+3*(i-1)+j-1)); if(matdet(M, 1)==0, print1(prime(k), ", ")))

Formula

a(n) = prime(A117345(n)).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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