A374067 -id:A374067 - cp's OEIS frontend

A374068 a(n) is the permanent of the symmetric Toeplitz matrix of order n whose element (i,j) equals the |i-j|-th prime or 0 if i = j.

1, 0, 4, 24, 529, 16100, 919037, 75568846, 9196890092, 1491628025318, 317579623173729, 86997150829931700, 29703399282858184713, 12512837775355494800500, 6397110844644502402189404, 3875565057688532269985283868, 2747710211567246171588232074225, 2265312860218073375019946448731300

Offset: 0

Stefano Spezia, Jun 27 2024

nonn

Conjecture: a(n) is the minimal permanent of an n X n symmetric Toeplitz matrix having 0 on the main diagonal and all the first n-1 primes off-diagonal. - Stefano Spezia, Jul 06 2024

			a(4) = 529:
  [0, 2, 3, 5]
  [2, 0, 2, 3]
  [3, 2, 0, 2]
  [5, 3, 2, 0]

Cf. A071079 (determinant), A085807, A306457, A318173.

Cf. A374067, A374069, A374070, A374071.

a[n_]:=Permanent[Table[If[i == j, 0, Prime[Abs[i - j]]], {i, 1, n}, {j, 1, n}]]; Join[{1},Array[a, 17]]

a(n) = matpermanent(matrix(n, n, i, j, if (i==j, 0, prime(abs(i-j))))); \\ Michel Marcus, Jun 28 2024

A374070 a(n) is the permanent of the symmetric Toeplitz matrix of order n whose element (i,j) equals the |i-j|-th composite or 0 if i = j.

Original entry on oeis.org

1, 0, 16, 192, 7056, 296928, 17353552, 1288517448, 123247560033, 14559205069230, 2068503986414344, 350413573991639400, 70216794936245622096, 16348540980271313405736, 4358673413318637872138056, 1324443244518891911978887758, 453726273130387432163560157389, 173630294056619179637594095141048

Offset: 0

Stefano Spezia, Jun 27 2024

nonn

			a(4) = 7056:
  [0, 4, 6, 8]
  [4, 0, 4, 6]
  [6, 4, 0, 4]
  [8, 6, 4, 0]

Cf. A071081 (determinant).

Cf. A374067, A374068, A374069, A374071.

Composite[n_Integer]:=FixedPoint[n + PrimePi[#] + 1 &, n + PrimePi[n] + 1]; a[n_]:=Permanent[Table[If[i == j, 0, Composite[Abs[i - j]]], {i, 1, n}, {j, 1, n}]]; Join[{1},Array[a,17]]

a(n) = my(composite(n)=my(k=-1); while(-n+n+=-k+k=primepi(n), ); n); matpermanent(matrix(n, n, i, j, if(i==j, 0, composite(abs(i-j))))); \\ Ruud H.G. van Tol, Jul 14 2024

from sympy import Matrix, composite
def A374070(n): return Matrix(n,n,[composite(abs(j-k)) if j!=k else 0 for j in range(n) for k in range(n)]).per() if n else 1 # Chai Wah Wu, Jul 01 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

Stefano Spezia, Jul 05 2024

			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]

Wikipedia, Toeplitz Matrix.

Cf. A071078, A374239.

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

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

Stefano Spezia, Jul 05 2024

			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]

Wikipedia, Toeplitz Matrix.

Cf. A071078, A374240.

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

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

Stefano Spezia, Jul 05 2024

			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]

Wikipedia, Toeplitz Matrix.

Cf. A071078, A374241.

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

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

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

A374069 a(n) is the permanent of the symmetric Toeplitz matrix of order n whose element (i,j) equals the |i-j|-th composite or 1 if i = j.

Original entry on oeis.org

1, 1, 17, 261, 8393, 356618, 20355656, 1498310848, 141920467648, 16632516446720, 2345863766165536, 394823892589979472, 78653652638945445776, 18216229760067802231488, 4833321599094565894295552, 1462259517864407783009737728, 498935238969900279377677930496, 190227655207141695023381769820864

Offset: 0

Stefano Spezia, Jun 27 2024

nonn

			a(4) = 8393:
  [1, 4, 6, 8]
  [4, 1, 4, 6]
  [6, 4, 1, 4]
  [8, 6, 4, 1]

Cf. A071080 (determinant).

Cf. A374067, A374068, A374070, A374071.

Composite[n_Integer]:=FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; a[n_] := Permanent[Table[If[i == j, 1, Composite[Abs[i - j]]], {i, 1, n}, {j, 1, n}]]; Join[{1},Array[a,17]]

c(n) = for(k=0, primepi(n), isprime(n++)&&k--); n; \\ A002808
a(n) = matpermanent(matrix(n, n, i, j, if (i==j, 1, c(abs(i-j))))); \\ Michel Marcus, Jun 27 2024

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

Stefano Spezia, Jul 05 2024

			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]

Wikipedia, Toeplitz Matrix.

Cf. A071078, A374242.

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

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]]

A374345 a(n) is the maximal permanent 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, 5, 59, 2454, 177998, 36960008, 7670953632, 2822399976144, 1061085324952592, 598646324654443008

Offset: 0

Stefano Spezia, Jul 05 2024

			a(5) = 177998:
  [1, 7, 5, 3, 2]
  [7, 1, 7, 5, 3]
  [5, 7, 1, 7, 5]
  [3, 5, 7, 1, 7]
  [2, 3, 5, 7, 1]

Wikipedia, Toeplitz Matrix.

Cf. A374067, A374278.

Cf. A374340, A374341, A374342, A374343, A374067 (minimal).

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

A374071 a(n) is the permanent of the Toeplitz matrix of order n whose element (i,j) equals the (i-j)-th composite number if i > j, (j-i)-th prime number if i < j, or 1 if i = j.

Original entry on oeis.org

1, 1, 9, 107, 2609, 98089, 5564610, 438180102, 46399705928, 6279673881161, 1060663766284535, 222840745939132105, 56798048066468972011, 17364018690978269373950, 6261448805827102522607660, 2624315396531837995006160020, 1263427401352418949898456181999, 693487403043958170112254851399169

Offset: 0

Stefano Spezia, Jun 27 2024

nonn

			a(4) = 2609:
  [1, 2, 3, 5]
  [4, 1, 2, 3]
  [6, 4, 1, 2]
  [8, 6, 4, 1]

Cf. A071082 (determinant).

Cf. A374067, A374068, A374069, A374070.

P,C:= selectremove(isprime,[$2..100]):
f:= proc(n) local i; uses LinearAlgebra;
  Permanent(ToeplitzMatrix([seq(C[i],i=n-1..1,-1),1,seq(P[i],i=1..n-1)]))
end proc:
map(f, [$0..20]); # Robert Israel, Jun 27 2024

Composite[n_Integer] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; a[n_]:= Permanent[Table[If[i == j, 1, If[i > j, Composite[i - j], Prime[j - i]]], {i, 1, n}, {j, 1, n}]]; Join[{1},Array[a, 17]]

A374140 a(n) is the permanent 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, 2, 11, 117, 2083, 55482, 2063149, 102176977, 6490667261, 514651043730, 49787897503031, 5771746960693493, 789652404867861919, 125885777192807718730, 23129357587464094132601, 4851600400570400272371009, 1152232847579194480216644249, 307579355879152834353840187554

Offset: 0

Stefano Spezia, Jun 28 2024

nonn

Conjecture: a(n) is the minimal permanent of an n X n symmetric Toeplitz matrix having 1 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal. - Stefano Spezia, Jul 05 2024

			a(4) = 117:
  [1, 1, 2, 3]
  [1, 1, 1, 2]
  [2, 1, 1, 1]
  [3, 2, 1, 1]

Cf. A085807, A374067, A374139 (determinant).

a[n_]:=Permanent[Table[If[i == j, 1, Abs[i - j]], {i, n}, {j, n}]]; Join[{1}, Array[a, 18]]

a(n) = matpermanent(matrix(n, n, i, j, if (i==j, 1, abs(i-j)))); \\ Michel Marcus, Jun 29 2024

from sympy import Matrix
def A374140(n): return Matrix(n,n,[abs(j-k) if j!=k else 1 for j in range(n) for k in range(n)]).per() if n else 1 # Chai Wah Wu, Jul 01 2024

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A374068 a(n) is the permanent of the symmetric Toeplitz matrix of order n whose element (i,j) equals the |i-j|-th prime or 0 if i = j.

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A374070 a(n) is the permanent of the symmetric Toeplitz matrix of order n whose element (i,j) equals the |i-j|-th composite or 0 if i = j.

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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.

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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.

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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.

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A374069 a(n) is the permanent of the symmetric Toeplitz matrix of order n whose element (i,j) equals the |i-j|-th composite or 1 if i = j.

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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.

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A374345 a(n) is the maximal permanent of an n X n symmetric Toeplitz matrix having 1 on the main diagonal and all the first n-1 primes off-diagonal.

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A374071 a(n) is the permanent of the Toeplitz matrix of order n whose element (i,j) equals the (i-j)-th composite number if i > j, (j-i)-th prime number if i < j, or 1 if i = j.

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