A219326 -id:A219326 - cp's OEIS frontend

A219324 Positive integers n that are equal to the determinant of the circulant matrix formed by the decimal digits of n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 247, 370, 378, 407, 481, 518, 592, 629, 1360, 3075, 26027, 26933, 45018, 69781, 80487, 154791, 1920261, 2137616, 2716713, 3100883, 3480140, 3934896, 4179451, 4830936, 5218958, 11955168, 80651025, 95738203, 257059332, 278945612, 456790123, 469135802, 493827160, 494376160

Offset: 1

Max Alekseyev, Nov 17 2012

Belukhov proved that if d is an odd divisor of p-1, then for integers q=(p^d-1)/((p-1)*d) and t such that (p-1)*(d-1)/2 < t < (p-1)*(d+1)/2 and gcd(t,d)=1, the number q*t equals the determinant of the circulant matrix formed by its base-p digits. For this sequence (where p=10), not every term can be obtained in this way.
If you rotate left (or take the absolute value of the determinant), then the sequence contains the following additional terms: 48, 1547, 123823, 289835, 23203827, ... (cf. A219326, A219327). - Robert G. Wilson v, Dec 12 2012
a(58) > 6*10^11. - Giovanni Resta, Dec 14 2012
See also A303260 for a different generalization: n X n circulant determinant having its base n+1 digits equal to a row. - M. F. Hasler, Apr 23 2018

			          | 2 4 7 |
247 = det | 7 2 4 |
          | 4 7 2 |

Giovanni Resta, Table of n, a(n) for n = 1..57 (first 47 terms from Robert G. Wilson v)
Max Alekseyev, Illustration for a(40) = 456790123
N. I. Belukhov, Solution to Problem 14.7 (in Russian), Matematicheskoe Prosveshchenie 15 (2011), pp. 241-244.
Wikipedia, Circulant matrix

Cf. A219325 (binary digits), A219326 (digits in reverse order), A219327 (absolute value of determinant), A306853 (permanent).

Cf. A303260.

f[n_] := Det[ NestList[ RotateRight@# &, IntegerDigits@ n, Floor[ Log10[n] + 1] - 1]]; k = 1; lst = {}; While[k < 1120000000, a = f@ k; If[a == k, AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Nov 20 2012 *)
Select[Range[53*10^5],Det[Table[RotateRight[IntegerDigits[#],d],{d,0,IntegerLength[ #]-1}]]==#&] (* The program generates the first 34 terms of the sequence. To generate more, increase the Range constant, but the program will take a long time to run. *) (* Harvey P. Dale, Jul 05 2021 *)

{ isA219324(n) = local(d,m,r); d=eval(Vec(Str(n))); m=#d; r=Mod(x,polcyclo(m)); prod(j=1,m,sum(i=1,m,d[i]*r^((i-1)*j)))==n }

from sympy import Matrix
A219324_list = []
for n in range(1,10**4):
    s = [int(d) for d in str(n)]
    m = len(s)
    if n == Matrix(m, m, lambda i, j: s[(i-j) % m]).det():
        A219324_list.append(n) # Chai Wah Wu, Oct 18 2021

A219327 Positive integers k that are equal to the absolute value of the determinant of the circulant matrix formed by the decimal digits of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 48, 247, 370, 378, 407, 481, 518, 592, 629, 1360, 1547, 3075, 26027, 26933, 45018, 69781, 80487, 123823, 154791, 289835, 1920261, 2137616, 2716713, 3100883, 3480140, 3934896, 4179451, 4830936, 5218958, 11955168, 23203827, 80651025, 95738203

Offset: 1

Max Alekseyev, Nov 17 2012

Contains A219324 and A219326 as subsequences.

Equal to the sequence defined by replacing circulant matrices with left circulant matrices. - Chai Wah Wu, Oct 18 2021

Robert G. Wilson v and Max Alekseyev, Table of n, a(n) for n = 1..63 (complete up to 10^10)

A219324, the main entry for this sequence, provides references and further details.

Cf. A219326.

from sympy import Matrix
A219327_list = []
for n in range(1,10**6):
    s = [int(d) for d in str(n)]
    m = len(s)
    if n == abs(Matrix(m, m, lambda i, j: s[(i-j) % m]).det()):
        A219327_list.append(n) # Chai Wah Wu, Oct 18 2021

a(53)-a(63) from Max Alekseyev, Feb 15 2013

A303261 Numbers having n digits in base n+1, and equal to the determinant of a circulant matrix based on these digits.

Original entry on oeis.org

1, 28, 35, 1936, 2761, 3421, 3732, 4043, 4354, 281048, 289820, 333293, 420239, 428752, 430686, 437554, 500380, 500888, 736600, 941578, 984377, 1027176, 1069975, 1112774, 1155573, 1662216, 1776201, 2087008, 5331625, 6825024, 7014400

Offset: 1

M. F. Hasler, Apr 25 2018

A subsequence of A303262, namely, the terms in row n which correspond to n-digit numbers in base n+1.

Cf. A303262, A303260, A219324, A219325, A219326, A219327.

for(n=1, 10, for(k=(n+1)^(n-1), (n+1)^n-1, d=Vec(digits(k, n+1)); abs(matdet(matrix(n, n, i, j, d[(j-i)%n+1])))==k&&print1(k", ")))

A303262 Table where row n lists numbers N equal to the determinant of an n X n circulant having as a row the base n+1 digits of N.

Original entry on oeis.org

1, 1, 1, 8, 9, 28, 35, 1, 65, 80, 91, 1, 44, 99, 550, 854, 1936, 2761, 3421, 3732, 4043, 4354, 1, 63, 65, 2527, 3311, 3969, 4095, 13545, 13889, 1, 128, 129, 145, 6066, 16384, 16385, 16512, 16513, 16641, 18560, 18577, 18669, 18705, 90738, 103759, 103965, 109220, 120142, 121920

Offset: 1

M. F. Hasler, Apr 23 2018

			The table starts
(n=1) 1,
(n=2) 1,
(n=3) 1, 8, 9, 28, 35,
(n=4) 1, 65, 80, 91,
(n=5) 1, 44, 99, 550, 854, 1936, 2761, 3421, 3732, 4043, 4354,
(n=6) 1, 63, 65, 2527, 3311, 3969, 4095, 13545, 13889,
(n=7) 1, 128, 129, 145, 6066, 16384, 16385, 16512, 16513, 16641, 18560, 18577, 18669, 18705, 90738, 103759, 103965, ...
For example, T(3,1) = 1 because the determinant of the circulant starting with [0, 0, 1] is 1. For the same reason each row starts with 1.
T(3,2) = 8 = 020[4] (digits in base 4) = det(circulant([0, 2, 0])).
T(3,5) = 35 = 203[4] = det(circulant([2, 0, 3])).

Cf. A303261, A303260, A219324, A219325, A219326, A219327.

for(n=1,7,for(k=1,(n+1)^n-1,d=Vec(digits(k,n+1),-n);abs(matdet(matrix(n,n,i,j,d[(j-i)%n+1])))==k&&print1(k",")))

A348428 Positive integers m that are equal to the determinant of the left circulant matrix formed by the decimal digits of m.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1547, 26027, 26933, 45018, 69781, 80487, 154791, 23203827, 257059332, 278945612, 456790123, 469135802, 493827160, 494376160, 506172839, 530864197, 543209876, 897163795, 1662971175, 2293668391, 3880266075, 6473710191

Offset: 1

Chai Wah Wu, Oct 18 2021

A left circulant matrix is also called a anti-circulant or (-1)-circulant matrix.
Subsequence of A219327.
Fixed points of A177894. - John Keith, Oct 24 2021

			           ⎡1  5  4  7⎤
1547 = det ⎢5  4  7  1⎥
           ⎢4  7  1  5⎥
           ⎣7  1  5  4⎦.

Cf. A219324, A219326, A219327.

Cf. A177894.

Select[Range[10^6], Equal[Det[NestList[RotateLeft, #2, #3 - 1]], #1] & @@ {#1, #2, Length[#2]} & @@ {#, IntegerDigits[#]} &] (* Michael De Vlieger, Oct 18 2021 *)

isok(m) = {my(d=digits(m), x); matdet(matrix(#d, #d, i, j, if (i==1, d[j], x = lift(Mod(j+i-1, #d)); if (!x, x += #d); d[x]))) == m;} \\ Michel Marcus, Oct 19 2021

from sympy import Matrix
A348428_list = []
for n in range(1,10**6):
    s = [int(d) for d in str(n)]
    m = len(s)
    if n == Matrix(m, m, lambda i, j: s[(i+j) % m]).det():
        A348428_list.append(n)

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A219324 Positive integers n that are equal to the determinant of the circulant matrix formed by the decimal digits of n.

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A219327 Positive integers k that are equal to the absolute value of the determinant of the circulant matrix formed by the decimal digits of k.

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A303261 Numbers having n digits in base n+1, and equal to the determinant of a circulant matrix based on these digits.

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A303262 Table where row n lists numbers N equal to the determinant of an n X n circulant having as a row the base n+1 digits of N.

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A348428 Positive integers m that are equal to the determinant of the left circulant matrix formed by the decimal digits of m.

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