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 12 results. Next

A374067 a(n) is the permanent of the symmetric Toeplitz matrix of order n whose element (i,j) equals the |i-j|-th prime or 1 if i = j.

Original entry on oeis.org

1, 1, 5, 42, 753, 22969, 1226225, 98413280, 11551199289, 1828335971613, 379823112871605, 102232301626742202, 34359550765856135217, 14289766516805617273497, 7224166042347461997365713, 4334493536305030883929928032, 3046742350470292308074313518937, 2492781304663024301187012794633153
Offset: 0

Views

Author

Stefano Spezia, Jun 27 2024

Keywords

Comments

Conjecture: a(n) is the minimal permanent of an n X n symmetric Toeplitz matrix having 1 on the main diagonal and all the first n-1 primes off-diagonal. - Stefano Spezia, Jul 08 2024

Examples

			a(4) = 753:
  [1, 2, 3, 5]
  [2, 1, 2, 3]
  [3, 2, 1, 2]
  [5, 3, 2, 1]

Crossrefs

Cf. A071078 (determinant), A306457, A318173.
Cf. A374068, A374069, A374070, A374071.

Programs

  • Mathematica
    a[n_]:=Permanent[Table[ If[i == j, 1, Prime[Abs[i - j]]], {i, 1, n}, {j, 1, n}]]; Join[{1},Array[a, 17]]
  • PARI
    a(n) = matpermanent(matrix(n, n, i, j, if (i==j, 1, prime(abs(i-j))))); \\ Michel Marcus, Jun 27 2024

A071079 Determinant of the n X n matrix whose element (i,j) equals the |i-j|-th prime or if i=j, 0.

Original entry on oeis.org

1, 0, -4, 24, -135, 696, -3577, 8738, -21120, 7182, 85995, 307692, -4739379, 72588316, 1042545852, 13023653868, -40433805531, -160245671048, 2415459163787, 80057252715814, -4733090488738644, 277504161503477090, -7200884073495803561, 97750528494490914120
Offset: 0

Views

Author

Robert G. Wilson v, May 26 2002

Keywords

Crossrefs

Cf. A071078, A071082, A071083.

Programs

  • Maple
    a:= n-> LinearAlgebra[Determinant](Matrix(n,
   (i, j)-> `if`(i=j, 0, ithprime(abs(i-j))))):
seq(a(n), n=0..24);  # Alois P. Heinz, Jun 27 2024
  • Mathematica
    f[n_] := Det[ Table[ If[ i == j, 0, Prime[ Abs[i - j]]], {i, 1, n}, {j, 1, n}]]; Table[ f[n], {n, 1, 20}]

Extensions

More terms from Sean A. Irvine, Jun 27 2024
a(0)=1 prepended by Alois P. Heinz, Jun 27 2024

A374139 a(n) is the determinant of the symmetric Toeplitz matrix of order n whose element (i,j) equals abs(i-j) or 1 if i = j.

Original entry on oeis.org

1, 1, 0, -1, 1, 3, 0, -3, 1, 5, 0, -5, 1, 7, 0, -7, 1, 9, 0, -9, 1, 11, 0, -11, 1, 13, 0, -13, 1, 15, 0, -15, 1, 17, 0, -17, 1, 19, 0, -19, 1, 21, 0, -21, 1, 23, 0, -23, 1, 25, 0, -25, 1, 27, 0, -27, 1, 29, 0, -29, 1, 31, 0, -31, 1, 33, 0, -33, 1, 35, 0, -35, 1, 37, 0, -37
Offset: 0

Views

Author

Stefano Spezia, Jun 28 2024

Keywords

Comments

A minor variant of A166445. - R. J. Mathar, Jul 01 2024

Examples

			a(4) = 1:
  [1, 1, 2, 3]
  [1, 1, 1, 2]
  [2, 1, 1, 1]
  [3, 2, 1, 1]

Links

Crossrefs

Cf. A056594, A071078, A085750, A374140 (permanent).

Programs

  • Mathematica
    a[n_]:=Det[Table[If[i == j, 1, Abs[i - j]], {i, n}, {j, n}]]; Join[{1}, Array[a, 75]]
  • PARI
    a(n) = matdet(matrix(n, n, i, j, if (i==j, 1, abs(i-j)))); \\ Michel Marcus, Jun 29 2024
  • Python
    from sympy import Matrix
def A374139(n): return Matrix(n,n,[abs(j-k) if j!=k else 1 for j in range(n) for k in range(n)]).det() # Chai Wah Wu, Jul 01 2024

Formula

G.f.: (1 + x^2 - x^3 + x^4)/((1 - x)*(1 + x^2)^2).
a(n) = a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) + a(n-5) for n > 4.
a(n) = (1 + A056594(n) + n*A056594(n+1))/2.
E.g.f.: (exp(x) + (1 + x)*cos(x))/2.
For a proof of the generating function and the recursion formula, see MathOverflow link. - Sela Fried, Jul 09 2024

A071082 Determinant of the n X n matrix whose element (i,j) equals the (i-j)-th composite number, (j-i)-th prime number, or 1 if i=j.

Original entry on oeis.org

1, -7, 39, -231, 1175, -6404, 34516, -194372, 914065, -4380707, 19511875, -48269825, 364029100, -2195115952, 13627012744, -115725814173, 792363218461, -5961225064275, 50261904138348, -425928565835370, 3704468293623774, -34926740161083290, 389473974875205556
Offset: 1

Views

Author

Robert G. Wilson v, May 26 2002

Keywords

Crossrefs

Cf. A071078, A071079.

Programs

  • Mathematica
    Composite[n_Integer] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; f[n_] := Det[ Table[ If[i == j, 1, If[i > j, Composite[i - j], Prime[j - i]]], {i, 1, n}, {j, 1, n}]]; Table[ f[n], {n, 1, 20}]

Extensions

More terms from Sean A. Irvine, Jun 27 2024

A071083 Determinant of the n X n matrix whose element (i,j) equals phi(|i-j|).

Original entry on oeis.org

0, -1, 2, -4, 6, -5, -84, 245, 924, -8816, -78556, -58667, 6779732, 10484275, -363899598, -6434306752, -13046891490, 20608641823, 1894093427008, -6177545952015, 18227184915788, -46023445768832, -8765268354146700, -1210879926442292743, -149657142827643535248
Offset: 1

Views

Author

Robert G. Wilson v, May 26 2002

Keywords

Crossrefs

Cf. A071078, A071079, A071082.

Programs

  • Mathematica
    f[n_] := Det[ Table[ EulerPhi[ Abs[i - j]], {i, 1, n}, {j, 1, n}]]; Table[ f[n], {n, 1, 22}]

Extensions

More terms from Sean A. Irvine, Jun 27 2024

A374340 a(n) is the minimal determinant of an n X n symmetric Toeplitz matrix having 1 on the main diagonal and all the first n-1 primes off-diagonal.

Original entry on oeis.org

1, 1, -3, 8, -21, -12167, -1708047, -116428560, -33081320935, -1098860747703, -579469550006151
Offset: 0

Views

Author

Stefano Spezia, Jul 05 2024

Keywords

Examples

			a(5) = -12167:
  [1, 2, 7, 3, 5]
  [2, 1, 2, 7, 3]
  [7, 2, 1, 2, 7]
  [3, 7, 2, 1, 2]
  [5, 3, 7, 2, 1]

Links

Crossrefs

Cf. A071078, A374239.
Cf. A374341 (maximal), A374342 (maximal absolute value), A374343 (minimal nonzero absolute value), A374067 (minimal permanent), A374345 (maximal permanent).

Programs

  • Mathematica
    a[n_]:=Min[Table[Det[ToeplitzMatrix[Join[{1},Part[Permutations[Prime[Range[n-1]]],i]]]],{i,(n-1)!}]]; Join[{1},Array[a,10]]

A374341 a(n) is the maximal determinant of an n X n symmetric Toeplitz matrix having 1 on the main diagonal and all the first n-1 primes off-diagonal.

Original entry on oeis.org

1, 1, -3, 15, 259, 1608, 1582152, 157042600, 11778545664, 3336975844504, 440384712302421
Offset: 0

Views

Author

Stefano Spezia, Jul 05 2024

Keywords

Examples

			a(5) = 1608:
  [1, 7, 2, 3, 5]
  [7, 1, 7, 2, 3]
  [2, 7, 1, 7, 2]
  [3, 2, 7, 1, 7]
  [5, 3, 2, 7, 1]

Links

Crossrefs

Cf. A071078, A374240.
Cf. A374340 (minimal), A374342 (maximal absolute value), A374343 (minimal nonzero absolute value), A374067 (minimal permanent), A374345 (maximal permanent).

Programs

  • Mathematica
    a[n_]:=Max[Table[Det[ToeplitzMatrix[Join[{1},Part[Permutations[Prime[Range[n-1]]],i]]]],{i,(n-1)!}]]; Join[{1},Array[a,10]]

A374342 a(n) is the maximal absolute value of the determinant of an n X n symmetric Toeplitz matrix having 1 on the main diagonal and all the first n-1 primes off-diagonal.

Original entry on oeis.org

1, 1, 3, 15, 259, 12167, 1708047, 157042600, 33081320935, 3336975844504, 579469550006151
Offset: 0

Views

Author

Stefano Spezia, Jul 05 2024

Keywords

Examples

			a(5) = 12167:
  [1, 2, 7, 3, 5]
  [2, 1, 2, 7, 3]
  [7, 2, 1, 2, 7]
  [3, 7, 2, 1, 2]
  [5, 3, 7, 2, 1]

Links

Crossrefs

Cf. A071078, A374241.
Cf. A374340 (minimal), A374341 (maximal), A374343 (minimal nonzero absolute value), A374067 (minimal permanent), A374345 (maximal permanent).

Programs

  • Mathematica
    a[n_]:=Max[Table[Abs[Det[ToeplitzMatrix[Join[{1},Part[Permutations[Prime[Range[n-1]]],i]]]]],{i,(n-1)!}]]; Join[{1},Array[a,10]]

Formula

a(n) = max(abs(A374340(n)),A374341(n)).

A374343 a(n) is the minimal absolute value of the determinant of a nonsingular n X n symmetric Toeplitz matrix having 1 on the main diagonal and all the first n-1 primes off-diagonal.

Original entry on oeis.org

1, 1, 3, 8, 7, 32, 81, 504, 327, 95
Offset: 0

Views

Author

Stefano Spezia, Jul 05 2024

Keywords

Examples

			a(5) = 32:
  [1, 3, 2, 5, 7]
  [3, 1, 3, 2, 5]
  [2, 3, 1, 3, 2]
  [5, 2, 3, 1, 3]
  [7, 5, 2, 3, 1]

Links

Crossrefs

Cf. A071078, A374242.
Cf. A374340 (minimal), A374341 (maximal), A374342 (maximal absolute value), A374067 (minimal permanent), A374345 (maximal permanent).

Programs

  • Mathematica
    a[n_]:=Min[Select[Table[Abs[Det[ToeplitzMatrix[Join[{1},Part[Permutations[Prime[Range[n-1]]],i]]]]],{i,(n-1)!}],Positive]]; Join[{1},Array[a,10]]

A071080 Determinant of the n X n matrix whose element (i,j) equals the |i-j|-th composite number, or 1 if i=j.

Original entry on oeis.org

1, -15, 125, -935, 6096, -38340, 240864, -1497584, 8611328, -49201152, 277473280, -1541996288, 7852493824, -39972516864, 195624648704, -789661486080, 3052709008384, -9659706075392, 30089357409792, -63825905935360, 63965499203712, -8296932715920, -1139418909751008
Offset: 1

Views

Author

Robert G. Wilson v, May 26 2002

Keywords

Links

Crossrefs

Cf. A071078, A071079, A071081, A071082, A071083.
Cf. A374069 (permanent).

Programs

  • Maple
    comps:= remove(isprime,[$4 .. 1000]):
f:= proc(n) local M;
  M:= Matrix(n,n,(i,j) -> `if`(i=j,1,comps[abs(i-j)]));
  LinearAlgebra:-Determinant(M)
end proc:
map(f, [$1..25]); # Robert Israel, Dec 03 2024
  • Mathematica
    Composite[n_Integer] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; f[n_] := Det[ Table[ If[i == j, 1, Composite[ Abs[i - j]]], {i, 1, n}, {j, 1, n}]]; Table[ f[n], {n, 1, 20}]

Extensions

a(21)-a(23) from Stefano Spezia, Jun 27 2024
Showing 1-10 of 12 results. Next

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