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 1-10 of 10 results.

A085000 Maximal determinant of an n X n matrix using the integers 1 to n^2.

Original entry on oeis.org

1, 10, 412, 40800, 6839492, 1865999570, 762150368499
Offset: 1

Views

Author

Robert G. Wilson v, Jun 16 2003

Keywords

Comments

Bounds for the next terms and the corresponding matrices are given by O. Gasper, H. Pfoertner and M. Sigg: 440960274696935 <= a(8) < 441077015225642, 346254605664223620 <= a(9) < 346335386150480625, 356944784622927045792 <= a(10) < 357017114947987625629. a(n) < sqrt(3*((n^5+n^4+n^3+n^2)/12)^n*(n^2+1)/(n+1)). - Hugo Pfoertner, Aug 15 2010
Improved lower bounds (private communication from Benjamin R. Buhrow, Dec 09 2019): a(8) >= 440970981670289, a(9) >= 346260899916111296. - Hugo Pfoertner, Jan 25 2021
Improved lower bound (private communication from Richard Gosiorovsky, Aug 18 2021): a(10) >= 356948996371054862392. - Hugo Pfoertner, Aug 24 2021

Examples

			The following 3 X 3 matrix is one of 36 whose determinant is 412 (there are also 36 whose determinant is -412):
   9 3 5
   4 8 1
   2 6 7
Results from a specially adapted hill-climbing algorithm strongly suggest that a(5) = 6839492. a(6) is at least 1862125166. Heuristically, a(n) is approximately 0.44*n^(2.06*n), suggesting that a(7) is close to 6.8*10^11. - Tim Paulden (timmy(AT)cantab.net), Sep 21 2003
a(5) confirmed by _Robert Israel_ and _Hugo Pfoertner_. A corresponding matrix is ((25 15 9 11 4) (7 24 14 3 17) (6 12 23 20 5) (10 13 2 22 19) (16 1 18 8 21) ). - _Hugo Pfoertner_, Sep 23 2003
a(6) found with FORTRAN program given at Pfoertner link. A corresponding matrix is ((36 24 21 17 5 8) ( 3 35 25 15 23 11) (13 7 34 16 10 31) (14 22 2 33 12 28) (20 4 19 29 32 6) (26 18 9 1 30 27) ). - _Hugo Pfoertner_, Sep 23 2003
a(7) is the determinant of the matrix ((46 42 15 2 27 24 18) (9 48 36 30 7 14 31) (39 11 44 34 13 29 5) (26 22 17 41 47 1 21) (20 8 40 6 33 23 45) (4 28 19 25 38 49 12) (32 16 3 37 10 35 43)). Although no proof for the optimality of a(7) is available, the results of an extensive computational search make the existence of a better solution extremely unlikely. A total of approximately 15 CPU years on SGI Origin 3000 and of 3.8 CPU years on SGI Altix 3000 computers was used for this result.

Links

Crossrefs

Cf. A088214, A088215, A088216, A088217, A088237, A180087 [upper bounds for a(n)], A180128, A221976, A301371, A301532, A301533, A309985, A325900, A350566.

Programs

  • Mathematica
    Needs["DiscreteMath`Combinatorica`"]; n=3; n2=n^2; dMax=0; mMax={}; p=Range[n2]; Do[m=Partition[p, n]; d=Det[m]; If[d>dMax, dMax=d; mMax=m]; p=NextPermutation[p], {k, n2!}]; {dMax, mMax} (* T. D. Noe *)
  • PARI
    vectomat(v)=my(n=sqrtint(#v));matrix(n,n,i,j,v[n*(i-1)+j])
a(n)=my(m,t,M); n*=n; for(k=0,(n-1)!-1, t=matdet(M=vectomat(numtoperm(n,k))); if(abs(t)>m, m=abs(t); print(t" "M)));m \\ Charles R Greathouse IV, Sep 13 2013

Extensions

a(4) from Marsac Laurent (jko(AT)rox0r.net), Sep 15 2003
a(6) from Hugo Pfoertner, Sep 23 2003
Entry edited by N. J. A. Sloane, Nov 22 2006, to remove some erroneous entries. Further edits Nov 25 2006.
a(7) from Hugo Pfoertner, Jan 22 2008

A350858 Minimal permanent of an n X n matrix whose elements are a permutation of the first n^2 prime numbers.

Original entry on oeis.org

1, 2, 29, 3664, 1820642, 2276752048, 5697057180536
Offset: 0

Views

Author

Stefano Spezia, Jan 19 2022

Keywords

Examples

			a(2) = 29:
    2    3
    5    7
a(3) = 3664:
    2    3    5
    7   13   19
   11   17   23

Crossrefs

Cf. A114533, A180128, A350565, A350859 (maximal).

Programs

  • Python
    from itertools import permutations
from sympy import Matrix
def A350858(n): return 1 if n == 0 else min(Matrix(n,n,p).per() for p in permutations(prime(m) for m in range(1,n**2+1))) # Chai Wah Wu, Jan 21 2022

Extensions

a(4)-a(6) from Hugo Pfoertner, Jan 21 2022

A350859 Maximal permanent of an n X n matrix whose elements are a permutation of the first n^2 prime numbers.

Original entry on oeis.org

1, 2, 41, 11868, 12124850, 25767879812, 101120963518528
Offset: 0

Views

Author

Stefano Spezia, Jan 19 2022

Keywords

Examples

			a(2) = 41:
   5    2
   3    7
a(3) = 11868:
  23    5    3
   2   13   19
   7   17   11

Crossrefs

Cf. A114533, A180128, A350566, A350858 (minimal).

Programs

  • Python
    from itertools import permutations
from sympy import Matrix
def A350859(n): return 1 if n == 0 else max(Matrix(n,n,p).per() for p in permutations(prime(m) for m in range(1,n**2+1))) # Chai Wah Wu, Jan 21 2022

Extensions

a(4)-a(6) from Hugo Pfoertner, Jan 21 2022

A351609 Maximal absolute value of the determinant of an n X n symmetric matrix using the integers 1 to n*(n + 1)/2.

Original entry on oeis.org

1, 1, 7, 152, 7113, 745285, 94974369
Offset: 0

Views

Author

Stefano Spezia, Feb 14 2022

Keywords

Comments

Upper bounds for the next terms can be found by considering all possibilities of choosing matrix entries on the diagonal and applying Gasper's determinant theorem (see references in A085000): a(7) <= 22475584128, a(8) <= 6634478203404, a(9) <= 2647044512044258. - Hugo Pfoertner, Feb 18 2022

Examples

			a(3) = 152:
   2    4    6
   4    5    1
   6    1    3
a(4) = 7113:
   2    6    8    9
   6    5   10    1
   8   10    3    4
   9    1    4    7

Crossrefs

Cf. A000217, A351147, A351148, A351153.
Cf. A085000, A180128, A350931, A350933, A350954, A350956.

Formula

a(n) = max(abs(A351147(n)), A351148(n)). - Hugo Pfoertner, Feb 16 2022

Extensions

a(5)-a(6) from Hugo Pfoertner, Feb 16 2022

A340923 4*a(n) is the maximum possible determinant of a 3 X 3 matrix whose entries are 9 consecutive primes starting with prime(n).

Original entry on oeis.org

1660, 2693, 3894, 5712, 7030, 9155, 10369, 11718, 14480, 16185, 18774, 20070, 22920, 24720, 23895, 26800, 31560, 39117, 43080, 43245, 42132, 38406, 41056, 48204, 66144, 69006, 86556, 98499, 99021, 88999, 77640, 87348, 86745, 89832, 92466, 95277, 98454, 84820
Offset: 1

Views

Author

Hugo Pfoertner, Jan 26 2021

Keywords

Comments

The entries of the matrix are arranged in such a way that the determinant of the matrix is maximized.

Examples

			a(1) = 1660 = A180128(3)/4 with the corresponding matrix shown in A180128.
a(2) = 2693: determinant (
  [13 29  7]
  [ 3 11 23]
  [19  5 17]) = 10772 = 4*2693.

Crossrefs

Cf. A117330, A180128, A340924, A340925.

Programs

  • Mathematica
    Table[Max[Det[Partition[#,3]]&/@Permutations[Prime[Range[n,n+8]]]],{n,40}]/4 (* Harvey P. Dale, Jul 21 2021 *)

A340924 8*a(n) is the maximum possible determinant of a 4 X 4 matrix whose entries are 16 consecutive primes starting with prime(n).

Original entry on oeis.org

608537, 837080, 1062261, 1335740, 1613011, 1834307, 2103606, 2369995, 2621808, 3072665, 3592140, 3891774, 4267302, 4412932, 4443915, 5039601, 5706864, 6673106, 7402050, 8535384, 9378963, 9989532, 10834096, 11530350, 11987568, 13560234, 14289963, 15119412, 15198123
Offset: 1

Views

Author

Hugo Pfoertner, Jan 26 2021

Keywords

Comments

The entries of the matrix are arranged in such a way that the determinant of the matrix is maximized.

Examples

			a(1) = 608537 = A180128(4)/8 with the corresponding matrix shown in A180128.
a(2) = 837080: determinant (
  [59 19 23  7]
  [11 53 37 13]
  [17  5 43 41]
  [29 31  3 47]) = 6696640 = 8*837080.

Crossrefs

Cf. A118799, A180128, A340923, A340925.

A340925 16*a(n) is the maximum possible determinant of a 5 X 5 matrix whose entries are 25 consecutive primes starting with prime(n).

Original entry on oeis.org

445934520, 527275650, 606375810, 668638620, 732258072, 860414368, 995563032, 1132837302, 1249798972, 1453587865, 1598993079, 1789976248, 2008319824, 2181193410, 2363922414, 2592209412, 2782039915, 3035727819, 3255326094, 3421333460, 3453338250, 3663999760, 4056944944
Offset: 2

Views

Author

Hugo Pfoertner, Jan 26 2021

Keywords

Comments

The entries of the matrix are arranged in such a way that the determinant of the matrix is maximized.
The special case of the first matrix with determinant A180128(5) = 5725998504 is excluded, since the prime number 2 prevents the otherwise existing divisibility of the determinant by 16.

Examples

			a(2) = 445934520: determinant(
  [73  53  3 79 23]
  [37 101 43  5 47]
  [19  41 89 71 13]
  [11  31 29 61 97]
  [83   7 67 17 59]) = 7134952320 = 16*445934520.

Crossrefs

Cf. A118815, A180128, A340923, A340924.

A180127 Upper bound for the determinant of an n X n matrix whose elements are a permutation of the first n^2 prime numbers.

Original entry on oeis.org

2, 32, 7414, 4993844, 5761178228, 11320943775475, 35966786849223443, 154715716383037989022, 1041732064414822689366009, 8436103376958505162325231670, 95816938885687281564299004113250, 1337411611273240103793149357629547975, 24089834168067078066162508828810807131186
Offset: 1

Views

Author

Hugo Pfoertner, Aug 12 2010

Keywords

Comments

a(n) is an upper bound for A180128(n).

Links

Crossrefs

Cf. A180128 [Maximal determinant of matrix with first n^2 primes], A085000 [Maximal determinant of matrix with elements 1, ..., n^2], A180087 [Upper bound for A085000], A007504 [Sum of first n primes], A024450 [Sum of first n squares of primes].

Programs

  • PARI
    a180127(n)={if(n<2,2, my(c=sum(k=1,n^2,prime(k))/n, d=sum(k=1,n^2,prime(k)^2)/n, t=(c^2-d)/(n-1)); floor(c*sqrt((d-t)^(n-1))))} \\ Hugo Pfoertner, Aug 27 2021

Formula

Let c = A007504(n^2)/n [(1/n)*sum of first n^2 primes]
and d = A024450(n^2)/n [(1/n)*sum of first n^2 squares of primes]
Then a(n) = floor(c*sqrt((d-t)^(n-1))) with t = (c^2-d)/(n-1).
log(a(n)) ~ (5*log(n) - log(3))*n/2 + n*log(log(n)). - Vaclav Kotesovec, Aug 28 2021

A364226 Triangle read by rows: T(n, k) is the number of n X n matrices of rank k using all the first n^2 prime numbers.

Original entry on oeis.org

1, 0, 24, 0, 216, 362664
Offset: 1

Views

Author

Stefano Spezia, Jul 15 2023

Keywords

Examples

			The triangle begins:
  1;
  0,  24;
  0, 216, 362664;
  ...

Crossrefs

Cf. A180128 (maximal determinant), A088020 (row sums), A350858 (minimal permanent), A350859 (maximal permanent), A364227 (right diagonal).
Cf. A364203 (with the integers in [n^2]).

A364227 a(n) is the number of n X n nonsingular matrices using all the first n^2 prime numbers.

Original entry on oeis.org

1, 24, 362664
Offset: 1

Views

Author

Stefano Spezia, Jul 15 2023

Keywords

Crossrefs

Right diagonal of A364226.
Cf. A180128 (maximal determinant), A350858 (minimal permanent), A350859 (maximal permanent).
Cf. A364206 (with the integers in [n^2]).
Showing 1-10 of 10 results.

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