A048954 -id:A048954 - cp's OEIS frontend

A086569 Product of the nonzero eigenvalues of the circulant matrix whose rows are formed by successively rotating a vector of binomial coefficients right. Generalization of A048954.

1, -3, 28, -375, 3751, -49392, 6835648, -1343091375, 364668913756, -210736858987743, 101832157445630503, -67043511427995648000, 487627751563388801409591, -4875797582053878382039400448, 58623274842128064372315087290368

Offset: 1

T. D. Noe, Jul 21 2003

In sequence A048954, a determinant of a circulant matrix, a(n) = 0 when 6 divides n. The determinant of a matrix can be interpreted as the signed volume of a simplex whose vertices are given by the rows of the matrix. For n a multiple of 6, the points form a lower dimensional simplex that has zero volume in n-space. However, the volume in n-2 space is positive and is given by the product of the nonzero eigenvalues.

			a(6) = -49392 because -1, -28, -28 and 63 are the four nonzero eigenvalues of the matrix {{1,6,15,20,15,6}, {6,1,6,15,20,15}, {15,6,1,6,15,20}, {20,15,6,1,6,15}, {15,20,15,6,1,6}, {6,15,20,15,6,1}}.

For references, see A086459

Cf. A048954, A086459 (circulant of powers of 2).

Table[x=Binomial[n, Range[0, n-1]]; m=Table[RotateRight[x, i-1], {i, n}]; e=Eigenvalues[m]; prod=1; Do[If[e[[i]]!=0, prod=prod*e[[i]]], {i, n}]; FullSimplify[prod], {n, 15}]

A215615 From Wendt's determinant compute sqrt(abs(A048954(n))/(2^n - 1)).

Original entry on oeis.org

1, 1, 2, 5, 11, 0, 232, 2295, 26714, 453871, 7053157, 0, 7715707299, 545539395584, 42297694603648, 4883188189089105, 531361846217471443, 0, 28649272821614715410221, 14214363393075742724609375, 7526219790642312236217153392, 5968603205606800870499639536231

Offset: 1

Jonathan Sondow, Aug 17 2012

nonn

E. Lehmer claimed, and J. S. Frame proved, that a(n) is an integer (Ribenboim 1999, p. 128).
The subsequence for even n is A129205.
See A048954 for additional comments, references, links, and cross-references.

P. Ribenboim, Fermat's Last Theorem for Amateurs, Springer-Verlag, NY, 1999, pp. 126, 136.

Cf. A048954, A129205, A215616.

w[n_] := Resultant[x^n - 1, (1 + x)^n - 1, x]; Table[ Sqrt[Abs[w[n]]/(2^n - 1)], {n, 25}]

a(n) = ((-1)^(n-1)*A048954(n)/(2^n - 1))^(1/2).

A215616 From Wendt's determinant compute (-A048954(2*n)/3)^(1/3).

Original entry on oeis.org

1, 5, 0, 765, 41261, 0, 1175731456, 804611664045, 0, 4133434158867578125, 36792671310208420147421, 0, 33666995638445382179718361163901, 3930778415673723952392425569428439040, 0, 637350736211692642266912139961455499346709367565

Offset: 1

Jonathan Sondow, Aug 17 2012

nonn

It is known that 3 divides A048954(2*n). It is conjectured that the quotient is a perfect cube.

See A048954 for additional comments, references, links, and cross-references.

Gerard P. Michon, Factorization of Wendt's Determinant(see Remarks and Conjectures).

Cf. A048954, A215615.

w[n_] := Resultant[x^n - 1, (1 + x)^n - 1, x]; Table[(-w[2 n]/3)^(1/3), {n, 19}]

a(n) = (-A048954(2*n)/3)^(1/3).

a(n) = 0 if and only if n is divisible by 3.

A129205 From Wendt's determinant compute (A048954(2*n)/(1-4^n))^(1/2).

Original entry on oeis.org

1, 5, 0, 2295, 453871, 0, 545539395584, 4883188189089105, 0, 14214363393075742724609375, 5968603205606800870499639536231, 0, 41302584753289717847206700750464023881130441

Offset: 1

Michael Somos, Apr 03 2007

nonn

P. Ribenboim, 13 Lectures on Fermat's last theorem, Springer-Verlag, NY, 1979, page 62. MR0551363 (81f:10023)

Cf. A048954, A215615, A215616.

{a(n)= if(n<1, 0, n*=2; sqrtint( matdet( matrix( n, n, i, j, binomial( n, (j-i)%n )))/ (1-2^n)))}

a(n)=0 if and only if n is divisible by 3.

a(n) = A215615(2*n). - Jonathan Sondow, Aug 17 2012

A336280 Number of consecutive primes of the form k*prime(n) + 1, starting with the least such prime A035095(n), that divides the Wendt determinant A048954(prime(n)).

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 2, 3, 2, 3, 1, 5, 6, 2, 6, 3, 3, 3, 1, 6, 3, 5, 5, 7, 5, 5, 6, 7, 4, 7, 5, 10, 10, 4, 4, 6, 10, 3, 4, 12, 11, 5, 7, 8, 7, 8, 11, 4, 4, 4, 14, 7, 11, 7, 13, 11, 13, 7, 18, 18, 6, 7, 17, 12, 9, 7, 9, 14, 12, 9, 16, 14, 11, 13, 10

Offset: 1

Frank M Jackson and Michael B Rees, Jul 15 2020

nonn

Michael B Rees has conjectured that:
1. for every prime p, the Wendt determinant Wendt(p) has all its prime factors that are greater than p of the form k*p + 1.
2. for every prime p = prime(n) and its corresponding Wendt determinant W(p) there exists a finite number of m consecutive primes (p_1,p_2,..,p_m) of the form  k*p + 1 that will divide Wendt(p) where p_1 is always the least prime of the form k*p + 1.
This sequence gives the value m for each p = prime(n).

			a(6) = 3 gives p = prime(6) = 13 and W(13) = 3^6*53^2*79^2*131^2*521^2*8191. The sequence of primes of the form q = k*13 + 1, starting with the least such prime 53 that divide W(11) is (53, 79, 131). The sequence has 3 terms.

Gerard P. Michon, Factorization of Wendt's Determinant (table for n=1 to 114).
Eric Weisstein's World of Mathematics, Circulant matrix .
Wikipedia, Circulant matrix .

Cf. A035095, A048954.

w[n_] := Module[{x}, Resultant[x^n-1, (1+x)^n-1, x]]; k[n_, m_] := Module[{p=Prime@n, q=0, lst={}}, Do[q++; While[! PrimeQ[p*q+1], q++]; AppendTo[lst, q], {m}]; lst];
lst1 = {}; Do[lst=k[n, 50]*Prime[n]+1; m = 1; Do[If[IntegerQ[w[Prime[n]]/lst[[m]]]&&m<=Length@lst, m++, Break[]], {Length@lst}]; AppendTo[lst1, m-1], {n, 1, 75}]; lst1

A335507 Index of the least Wendt determinant (A048954) divisible by prime(n).

Original entry on oeis.org

3, 2, 4, 3, 5, 28, 8, 9, 11, 7, 5, 9, 20, 14, 23, 13, 29, 15, 11, 35, 9, 13, 41, 11, 32, 25, 17, 53, 27, 28, 7, 13, 17, 23, 37, 15, 39, 27, 83, 43, 89, 45, 19, 32, 28, 11, 21, 37, 113, 19, 29, 34, 40, 25, 16, 131, 67, 15, 69, 35, 47, 73, 17, 31, 39, 79, 33, 21, 173, 29, 32, 179

Offset: 1

Frank M Jackson and Michael B Rees, Jun 11 2020

nonn

It has been conjectured by Michael B Rees that there exists for every prime a Wendt determinant divisible by that prime. However the conjecture has been proved for all prime divisors equivalent to -1 (mod 6) - (see Lehmer link below).

			a(5) = 5 because Wendt(5) = 3751 = 11^2*131. It is divisible by prime(5) = 11 and Wendt(5) is the least Wendt determinant divisible by 11.

Charles Helou, On Wendt's Determinant, Math. Comp., 66 (1997) No. 219, 1341-1346.
Emma Lehmer, On a Resultant Connected with Fermat's Last Theorem, Bull. Amer. Math. Soc. 41 (1935), 864-867.
Eric Weisstein's World of Mathematics, Circulant matrix.
Wikipedia, Circulant matrix.

Cf. A048954.

Wendt[n_]:=Module[{x},Resultant[x^n-1,(1+x)^n-1,x]];
findW[n_]:= Module[{m=1},While[!IntegerQ[Wendt[m]/n]||Mod[m,6]==0,m++];m];
Table[findW[Prime[n]],{n,1,100}]

A335970 Index of the least Wendt determinant (A048954(k)) that is divisible by the least prime of the form k*prime(n) + 1.

Original entry on oeis.org

3, 10, 26, 28, 32, 46, 38, 58, 44, 110, 22, 88, 122, 70, 134, 44, 164, 70, 106, 212, 94, 70, 146, 128, 208, 62, 142, 116, 310, 56, 94, 212, 86, 280, 320, 262, 316, 82, 110, 122, 182, 160, 362, 142, 284, 280, 340, 112, 56, 64, 254, 308, 250, 368, 104, 272, 242, 292, 226

Offset: 1

Frank M Jackson and Michael B Rees, Jul 03 2020

nonn

It has been conjectured by Michael B Rees that for every prime p there exists a Wendt determinant index j such that for all j < k primes of the form j*p + 1 will not divide Wendt(j). This sequence gives the least index k such that Wendt(k) is divisible by the least prime of the form k*p + 1 for each prime p = prime(n).

			a(2) = 10 gives k = 10, Wendt(10) = -210736858987743 = -1*3*11^9*31^3 and p = prime(2) = 3. Hence with k = 10 and p = 3, Wendt(k) is the least Wendt determinant divisible by the least prime of the form p*k + 1.

Gerard P. Michon, Factorization of Wendt's Determinant (table for n=1 to 114).
Eric Weisstein's World of Mathematics, Circulant matrix.
Wikipedia, Circulant matrix.

Cf. A048954.

w[n_] := Module[{x}, Resultant[x^n-1, (1+x)^n-1, x]]; lst = {}; Do[q=1; While[Mod[q, 6]==0||!PrimeQ[r=1+Prime[n]*q]||!IntegerQ[w[q]/r], q++]; AppendTo[lst, q], {n, 1, 60}]; lst

A336688 Primes p such that the Wendt determinant A048954(p) has prime factors less than p.

Original entry on oeis.org

3, 7, 13, 31, 73, 127, 307, 331, 757

Offset: 1

Frank M Jackson and Michael B Rees, Jul 31 2020

Michael B Rees has conjectured that for all primes p, each fully exponentiated prime factor less than p that divides the Wendt determinant W(p), if it exists, is of the form k*p + 1.
This sequence identifies the prime index p of Wendt determinants W(p) that have prime factors less than p.
These prime indices appear to be a subset of the lucky primes A031157.

			a(3) = 13. The Wendt determinant with a prime index p = 13 has prime factors less than p. W(13) = 3^6*53^2*79^2*131^2*521^2*8191 and 3^6 = 729 is of the form k*13 + 1. It is the 3rd occurrence of such a determinant.

Gerard P. Michon, Factorization of Wendt's Determinant (table for n=1 to 114).
Eric Weisstein's World of Mathematics, Circulant matrix.
Wikipedia, Circulant matrix.

Cf. A048954, A336280.

w[n_] := Resultant[x^n-1, (1+x)^n-1, x]; getp[n_] := Module[{W=w[n], lst=Table[Prime[m], {m, 1, PrimePi[n]}], lst1={}, j, k, l}, Do[j=1; While[W>0&&IntegerQ[W/lst[[l]]^j], j++]; If[j-1>0, AppendTo[lst1, {lst[[l]], j-1}]], {l, 1, Length@lst}]; Join[{n}, lst1]]; lst = {}; Do[lst1=getp[Prime[n]]; If[Length@lst1>1, AppendTo[lst, lst1[[1]]]], {n, 1, PrimePi[331]}]; lst

a(9) from Jinyuan Wang, Sep 04 2020

A086459 Determinant of the circulant matrix whose rows are formed by successively rotating the vector (1, 2, 4, 8, ..., 2^(n-1)) right.

Original entry on oeis.org

1, -3, 49, -3375, 923521, -992436543, 4195872914689, -70110209207109375, 4649081944211090042881, -1227102111503512992112190463, 1291749870339606615892191271170049, -5429914198235566686555216227881787109375

Offset: 1

T. D. Noe, Jul 21 2003

Note that if the rows are rotated left instead of right, the sign of the terms for which n = 0 or 3 (mod 4) is reversed. The n eigenvalues of these circulant matrices lie on the circle of radius 2(2^n - 1)/3 centered at x = (2^n - 1)/3, y = 0. This sequence can be generalized to bases other than 2 and similar results are true.

			a(3) = determinant of the matrix ((1,2,4),(4,1,2),(2,4,1)) = 49. [Corrected by _T. D. Noe_, Jan 22 2008]

Richard Bellman, Introduction to Matrix Analysis, Second Edition, SIAM, 1970, pp. 242-3.
Philip J. Davis, Circulant Matrices, Second Edition, Chelsea, 1994.

Eric Weisstein's World of Mathematics, Circulant Matrix

Cf. A048954 (circulant of binomial coefficients), A052182 (circulant of natural numbers), A066933 (circulant of prime numbers).

Cf. A180602 (unsigned, offset 0). [Paul D. Hanna, Sep 11 2010]

restart:with (combinat):a:=n->mul(-stirling2(n,2), j=3..n): seq(a(n), n=2..19); # Zerinvary Lajos, Jan 01 2009

Table[x=2^Range[0, n-1]; m=Table[RotateRight[x, i-1], {i, n}]; Det[m], {n, 12}]

a(n) = (-2^n + 1)^(n-1).

See formulas in A180602, an unsigned version of this sequence with offset 0. [Paul D. Hanna, Sep 11 2010]

A118713 a(n) = determinant of n X n circulant matrix whose first row is A001358(1), A001358(2), ..., A001358(n) where A001358(n) = n-th semiprime.

Original entry on oeis.org

4, -20, 361, -3567, 218053, -3455872, 736439027, -16245418225, 1519211613654, -37662452460912, 20199655476042865, -643524421698841536, 46513669467992431114, -3754367220494585505280, 277686193779526116536293, -123973821931125256333959105, 20103033234038999233385180658

Offset: 1

Jonathan Vos Post, May 20 2006

Semiprime analog of A066933 Circulant of prime numbers. a(n) alternates in sign. A048954 Wendt determinant of n-th circulant matrix C(n). A052182 Circulant of natural numbers. A086459 Circulant of powers of 2.

			a(2) = -20 = determinant
|4,6|
|6,4|.
a(3) = 361 = 19^2 = determinant
|4,6,9|
|9,4,6|
|6,9,4|.

Eric Weisstein's World of Mathematics, Circulant Matrix.

Cf. A001358, A048954, A052182, A066933, A086459, A086569.

A118713 := proc(n)
    local C,r,c ;
    C := Matrix(1..n,1..n) ;
    for r from 1 to n do
    for c from 1 to n do
        C[r,c] := A001358(1+((c-r) mod n)) ;
    end do:
    end do:
    LinearAlgebra[Determinant](C) ;
end proc:
seq(A118713(n),n=1..13) ;

nmax = 13;
sp = Select[Range[3 nmax], PrimeOmega[#] == 2&];
a[n_] := Module[{M}, M[1] = sp[[1 ;; n]];
   M[k_] := M[k] = RotateRight[M[k - 1]];
   Det[Table[M[k], {k, 1, n}]]];
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Feb 16 2023 *)

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Aug 23 2007

cp's OEIS Frontend

A086569 Product of the nonzero eigenvalues of the circulant matrix whose rows are formed by successively rotating a vector of binomial coefficients right. Generalization of A048954.

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A215615 From Wendt's determinant compute sqrt(abs(A048954(n))/(2^n - 1)).

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A215616 From Wendt's determinant compute (-A048954(2*n)/3)^(1/3).

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A129205 From Wendt's determinant compute (A048954(2*n)/(1-4^n))^(1/2).

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A336280 Number of consecutive primes of the form k*prime(n) + 1, starting with the least such prime A035095(n), that divides the Wendt determinant A048954(prime(n)).

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A335507 Index of the least Wendt determinant (A048954) divisible by prime(n).

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A335970 Index of the least Wendt determinant (A048954(k)) that is divisible by the least prime of the form k*prime(n) + 1.

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A336688 Primes p such that the Wendt determinant A048954(p) has prime factors less than p.

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