A069191 -id:A069191 - cp's OEIS frontend

A228548 Determinant of the n X n matrix with (i,j)-entry equal to A008683(i+j-1) for all i,j = 1..n.

1, -2, 3, 3, -7, -5, 12, -19, -52, -52, -20, 73, -919, 6209, 2206, -1869, -8835, -4021, 23202, -122489, -174347, 1106682, 1114088, 388318, -7528057, 55753005, 81020413, -530178192, -6348221604, 101952770365, -371734984964, -16091176203501, 90823940064758, 163339092651834, -3480231557696967

Offset: 1

Zhi-Wei Sun, Aug 25 2013

sign

Conjecture: a(n) is always nonzero. Moreover, |a(n)|^(1/n) tends to infinity.

We have verified that a(n) is nonzero for all n = 1..500.

			a(1) = 1 since Moebius(1+1-1) = 1.

Zhi-Wei Sun, Table of n, a(n) for n = 1..200

Cf. A008683, A228549, A069191, A228552.

a[n_]:=a[n]=Det[Table[MoebiusMu[i+j-1],{i,1,n},{j,1,n}]]
Table[a[n],{n,1,10}]

a(n) = matdet(matrix(n, n, i, j, moebius(i+j-1))); \\ Michel Marcus, Apr 14 2023

A228549 Determinant of the n X n matrix with (i,j)-entry equal to 1 or 0 according as i + j is squarefree or not.

Original entry on oeis.org

1, -1, -2, 3, 2, -1, -5, 25, -20, -16, -7, 23, -39, 25, 309, 455, 901, -2168, 920, 1047, 2302, -1536, 3272, 161162, -289493, -529263, 1418949, -1562590, -579999, 15145, 5844471, 5244416, -65749104, 176617384, 2077411081, 7381004327, 10211507191, -10333531933, 7086785521, 8502336136

Offset: 1

Zhi-Wei Sun, Aug 25 2013

sign

Conjecture: a(n) is always nonzero. Moreover, |a(n)|^(1/n) tends to infinity.

We have verified that a(n) is nonzero for all n <= 500.

			a(1) = 1 since 1 + 1 = 2 is squarefree.

Zhi-Wei Sun, Table of n, a(n) for n = 1..200

Cf. A005117, A228548, A069191, A228552.

a[n_]:=a[n]=Det[Table[If[SquareFreeQ[i+j]==True,1,0],{i,1,n},{j,1,n}]]
Table[a[n],{n,1,20}]
Table[With[{nn=x},Det[Table[If[SquareFreeQ[i+j],1,0],{i,nn},{j,nn}]]],{x,40}] (* Harvey P. Dale, Apr 02 2023 *)

A071524 Determinant of n X n matrix defined by m(i,j)=1 if i^2+j^2 is a prime, m(i,j)=0 otherwise.

Original entry on oeis.org

1, -1, -1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 16, 25, -25, -100, 1, 81, -16, -36, 0, 1764, -3136, -196, 324, 16, -225, -1764, 1521, 9, -3969, -4356, 4761, 9, -1225, -19881, 5041, 156816, -312481, -167281, 219024, 3600, -186624, -158404, 5541316, 3020644, -19554084, -1350244, 198810000

Offset: 1

Benoit Cloitre, Jun 02 2002

Terms are also perfect squares.
Conjecture: a(n) = 0 for no n > 28. - Zhi-Wei Sun, Aug 26 2013
General conjecture: Let m be any nonnegative integer, and let a(m,n) be the n X n determinant with (i,j)-entry equal to 1 or 0 according as i^{2^m}+j^{2^m} is prime or not. Then a(m,n) is nonzero for large n. (It can be proved that (-1)^(n*(n-1)/2)*a(m,n) is always a square, see the comments in A228591.) - Zhi-Wei Sun, Aug 26-27 2013

Cf. A069191, A228591, A228552, A228557, A228559, A228561, A228574, A228578.

a[n_]:=a[n]=Det[Table[If[PrimeQ[i^2+j^2]==True, 1, 0], {i, 1, n}, {j, 1, n}]]; Table[a[n], {n, 1, 30}] (* Zhi-Wei Sun, Aug 26 2013 *)
Table[Det[Table[If[PrimeQ[a^2+b^2],1,0],{a,n},{b,n}]],{n,60}] (* Harvey P. Dale, May 31 2019 *)

for(n=1,60,print1(((matdet(matrix(n,n,i,j,isprime(i^2+j^2))))),","))

A228624 Determinant of the n X n matrix with (i,j)-entry equal to 1 or 0 according as i + j is a square or not.

Original entry on oeis.org

0, 0, -1, 0, 1, 0, 0, 1, 1, 1, 0, -1, 1, 0, 0, -1, 2, 3, -3, -1, 0, 1, -1, -2, -5, 13, -7, -7, -6, 1, 8, -1, -17, 25, 13, -12, 11, 12, -11, -12, -4, 1, 1, -66, -60, -26, -13, 40, -67, -1, 82, 81, -49, -32, 68, 103, -222, 503, -39, -134

Offset: 1

Zhi-Wei Sun, Aug 28 2013

sign

Conjecture: a(n) is nonzero for any n > 21.
Zhi-Wei Sun also made the following similar conjecture:
  Let A(n) be the n X n determinant with (i,j)-entry equal to 1 or 0 according as i + j is a cube or not. Then A(n) is nonzero for any n > 176.

			a(1) = 0 since 1 + 1 = 2 is not a square.

Zhi-Wei Sun, Table of n, a(n) for n = 1..400

Cf. A000290, A069191, A228591, A228557, A228559, A228615, A228616, A228623.

SQ[n_]:=IntegerQ[Sqrt[n]]
a[n_]:=Det[Table[If[SQ[i+j]==True,1,0],{i,1,n},{j,1,n}]]
Table[a[n],{n,1,30}]

a(n)=matdet(matrix(n,n,i,j,issquare(i+j))) \\ Ralf Stephan, Sep 17 2013

A071063 Determinant of n X n matrix defined by m(i,j) = 0 if i+j is a prime, m(i,j) = 1 otherwise.

Original entry on oeis.org

0, 0, -1, 0, 1, 0, -9, -8, 0, 0, 0, 0, 0, 0, 0, -8, 9, 14, -71, -310, 281, 2000, -8004, -9200, 8836, 720, -409, -2710, 67766, 110501, -1117396, -4130160, 381136, 91920, -111376, -36080, 144420, 555581, -311814, -1831958, 1876689, -1648, -3584425, 4768308, 1971637204, 53664688220

Offset: 1

Santi Spadaro, May 26 2002

sign

Let h(i,j) be the matrix defined in A069191, then a(n)=((-1)^n)*Det(h(i,j)-J), where J is the n X n matrix with only 1's as its elements.

Cf. A069191.

a[n_] := Det[Table[If[PrimeQ[i + j], 0, 1], {i, 1, n}, {j, 1, n}]] Table[a[n], {n, 1, 50}]

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A228548 Determinant of the n X n matrix with (i,j)-entry equal to A008683(i+j-1) for all i,j = 1..n.

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A228549 Determinant of the n X n matrix with (i,j)-entry equal to 1 or 0 according as i + j is squarefree or not.

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A071524 Determinant of n X n matrix defined by m(i,j)=1 if i^2+j^2 is a prime, m(i,j)=0 otherwise.

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A228624 Determinant of the n X n matrix with (i,j)-entry equal to 1 or 0 according as i + j is a square or not.

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A071063 Determinant of n X n matrix defined by m(i,j) = 0 if i+j is a prime, m(i,j) = 1 otherwise.

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