A071078
Determinant of the n X n matrix whose element (i,j) equals the |i-j|-th prime or if i=j, 1.
Original entry on oeis.org
1, 1, -3, 8, -21, 45, -81, -504, -327, -95, -27, 3744, -514125, 42577129, -594343521, 6384803528, 55828902985, 222653960625, -24898185972135, 370012327363008, -5108693773098321, 66203904684406485, -696661740301346521, 5010137735370840432, -22341106332701627463
Offset: 0
-
f[n_] := Det[ Table[ If[ i == j, 1, Prime[ Abs[i - j]]], {i, 1, n}, {j, 1, n}]]; Table[ f[n], {n, 1, 21}]
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.
Original entry on oeis.org
1, 0, 4, 24, 529, 16100, 919037, 75568846, 9196890092, 1491628025318, 317579623173729, 86997150829931700, 29703399282858184713, 12512837775355494800500, 6397110844644502402189404, 3875565057688532269985283868, 2747710211567246171588232074225, 2265312860218073375019946448731300
Offset: 0
a(4) = 529:
[0, 2, 3, 5]
[2, 0, 2, 3]
[3, 2, 0, 2]
[5, 3, 2, 0]
-
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
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
-
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}]
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
-
f[n_] := Det[ Table[ EulerPhi[ Abs[i - j]], {i, 1, n}, {j, 1, n}]]; Table[ f[n], {n, 1, 22}]
A374386
a(n) is the minimal determinant of an n X n symmetric Toeplitz matrix having 0 on the main diagonal and all the first n-1 primes off-diagonal.
Original entry on oeis.org
1, 0, -4, 24, -324, -15164, -1453072, -161765904, -37905894000, -1376219654680, -718058901423168, -163742479201610036
Offset: 0
a(5) = -15164:
[0, 2, 7, 3, 5]
[2, 0, 2, 7, 3]
[7, 2, 0, 2, 7]
[3, 7, 2, 0, 2]
[5, 3, 7, 2, 0]
-
a[n_]:=Min[Table[Det[ToeplitzMatrix[Join[{0},Part[Permutations[Prime[Range[n-1]]],i]]]],{i,(n-1)!}]]; Join[{1},Array[a,10]]
A374387
a(n) is the maximal determinant of an n X n symmetric Toeplitz matrix having 0 on the main diagonal and all the first n-1 primes off-diagonal.
Original entry on oeis.org
1, 0, -4, 36, 129, 3340, 2287607, 162104000, 16943055268, 4059346088384, 474967482901952, 221630954408019520
Offset: 0
a(5) = 3340:
[0, 5, 7, 3, 2]
[5, 0, 5, 7, 3]
[7, 5, 0, 5, 7]
[3, 7, 5, 0, 5]
[2, 3, 7, 5, 0]
-
a[n_]:=Max[Table[Det[ToeplitzMatrix[Join[{0},Part[Permutations[Prime[Range[n-1]]],i]]]],{i,(n-1)!}]]; Join[{1},Array[a,10]]
A374388
a(n) is the maximal absolute value of the determinant of an n X n symmetric Toeplitz matrix having 0 on the main diagonal and all the first n-1 primes off-diagonal.
Original entry on oeis.org
1, 0, 4, 36, 324, 15164, 2287607, 162104000, 37905894000, 4059346088384, 718058901423168, 221630954408019520
Offset: 0
a(5) = 15164:
[0, 2, 7, 3, 5]
[2, 0, 2, 7, 3]
[7, 2, 0, 2, 7]
[3, 7, 2, 0, 2]
[5, 3, 7, 2, 0]
-
a[n_]:=Max[Table[Abs[Det[ToeplitzMatrix[Join[{0},Part[Permutations[Prime[Range[n-1]]],i]]]]],{i,(n-1)!}]]; 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
-
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
-
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}]
A071081
Determinant of the n X n matrix whose element (i,j) equals the |i-j|-th composite number, or 0 if i=j.
Original entry on oeis.org
1, 0, -16, 192, -1904, 16416, -134608, 1102920, -8971103, 69262338, -527129920, 4002967800, -30263030000, 218133853800, -1565386817920, 11130108480678, -75244171093875, 496516351214832, -3261752198331472, 21401161780748720, -140093238345715827, 914525302322457472
Offset: 0
-
comps:= remove(isprime,[$4 .. 11000]):
f:= proc(n) local M;
M:= Matrix(n,n,(i,j) -> `if`(i=j,0,comps[abs(i-j)]));
LinearAlgebra:-Determinant(M)
end proc:
f(0):= 1:
map(f, [$0..25]); # Robert Israel, Dec 02 2024
-
Composite[n_Integer] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; f[n_] := Det[ Table[ If[i == j, 0, Composite[ Abs[i - j]]], {i, 1, n}, {j, 1, n}]]; Table[ f[n], {n, 1, 20}]
-
a(n) = my(composite(n)=my(k=-1); while(-n+n+=-k+k=primepi(n), ); n); matdet(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 A071081(n): return Matrix(n,n,[composite(abs(j-k)) if j!=k else 0 for j in range(n) for k in range(n)]).det() # Chai Wah Wu, Jul 01 2024
A381514
a(n) is the hafnian of a symmetric Toeplitz matrix of order 2*n whose off-diagonal element (i,j) equals the |i-j|-th prime.
Original entry on oeis.org
1, 2, 23, 899, 85072, 15120411, 4439935299, 1989537541918, 1264044973158281, 1090056235155152713, 1227540523199054294506
Offset: 0
a(2) = 23 because the hafnian of
[d 2 3 5]
[2 d 2 3]
[3 2 d 2]
[5 3 2 d]
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 2*2 + 3*3 + 5*2 = 23. Here d denotes the generic element on the main diagonal of the matrix from which the hafnian does not depend.
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M[i_, j_]:=Prime[Abs[i-j]]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i]], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 5, 0]
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