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
a(4) = 753:
[1, 2, 3, 5]
[2, 1, 2, 3]
[3, 2, 1, 2]
[5, 3, 2, 1]
-
a[n_]:=Permanent[Table[ If[i == j, 1, 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, 1, prime(abs(i-j))))); \\ Michel Marcus, Jun 27 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
a(4) = 7056:
[0, 4, 6, 8]
[4, 0, 4, 6]
[6, 4, 0, 4]
[8, 6, 4, 0]
-
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
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
a(4) = 8393:
[1, 4, 6, 8]
[4, 1, 4, 6]
[6, 4, 1, 4]
[8, 6, 4, 1]
-
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]]
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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
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]]
A374389
a(n) is the minimal absolute value of the determinant of a nonsingular 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
4, 24, 116, 192, 1079, 664, 720, 216
Offset: 2
a(5) = 192:
[0, 5, 3, 2, 7]
[5, 0, 5, 3, 2]
[3, 5, 0, 5, 3]
[2, 3, 5, 0, 5]
[7, 2, 3, 5, 0]
-
a[n_]:=Min[Select[Table[Abs[Det[ToeplitzMatrix[Join[{0},Part[Permutations[Prime[Range[n-1]]],i]]]]],{i,(n-1)!}],Positive]]; Array[a,8,2]
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
a(4) = 2609:
[1, 2, 3, 5]
[4, 1, 2, 3]
[6, 4, 1, 2]
[8, 6, 4, 1]
-
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]]
A374390
a(n) is the maximal permanent 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, 1936, 144260, 31972988, 6800311204, 2560967581304, 975834087080060, 557171087172087364
Offset: 0
a(5) = 144260:
[0, 7, 5, 3, 2]
[7, 0, 7, 5, 3]
[5, 7, 0, 7, 5]
[3, 5, 7, 0, 7]
[2, 3, 5, 7, 0]
-
a[n_]:=Max[Table[Permanent[ToeplitzMatrix[Join[{0},Part[Permutations[Prime[Range[n-1]]],i]]]],{i,(n-1)!}]]; Join[{1},Array[a,10]]
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]
Showing 1-10 of 10 results.
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