A219325 -id:A219325 - 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

A306595 Determinant of the circulant matrix whose first column corresponds to the binary digits of n.

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 2, 0, 1, 0, 0, 3, 0, -3, 3, 0, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 0, 1, 0, 4, 0, 0, -9, 9, 0, 4, 9, 0, 8, 9, 0, 8, 5, 0, 0, 9, 0, -9, -8, 0, -5, 0, 0, 8, 5, 0, -5, 5, 0, 1, 2, 2, 3, 2, 24, 24, 4, 2, 3, 3, 32, 3, 4, 32, 5, 2, 24, 3

Offset: 0

Rémy Sigrist, Feb 27 2019

This sequence is the binary variant of A177894.
From Robert Israel, Mar 05 2019: (Start)
a(n) is divisible by A000120(n).
If A070939(n) is even then n is divisible by A000120(n)*A065359(n). (End)

			For n = 13:
- the binary representation of 13 is "1101",
- the corresponding circulant matrix is:
    [1 1 0 1]
    [1 1 1 0]
    [0 1 1 1]
    [1 0 1 1]
- its determinant is -3,
- hence a(13) = -3.

Robert Israel, Table of n, a(n) for n = 0..10000
Wikipedia, Circulant matrix
Index entries for sequences related to binary expansion of n

Cf. A000120, A065359, A070939, A121016, A177894, A219325, A306714.

a:= n-> `if`(n=1, 1, (l-> LinearAlgebra[Determinant](Matrix(nops(l),
       shape=Circulant[l[-i]$i=1..nops(l)])))(convert(n, base, 2))):
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 05 2019

a(n) = my (d=if (n, binary(n), [0])); my (m=matrix(#d, #d, i,j, d[1+(i-j)%#d])); return (matdet(m))

a(A121016(n)) = 0 for any n > 0.
a(2^k) = 1 for any k >= 0.
a(A219325(n)) = A219325(n) for any n > 0.

A219357 a(n) = smallest number greater than n, equal to the determinant of the circulant matrix formed by its base-n digits.

Original entry on oeis.org

17298, 1352, 28, 28, 320, 81, 133, 104, 247, 126, 1273, 252, 793, 473, 520, 980, 832, 513, 468, 5792, 684, 1738, 2511, 684, 1520, 14711, 7588, 938, 3857, 2275, 4680, 13392, 5184, 1648, 10535, 1820, 9143, 8473, 3843, 21880, 11609, 3843

Offset: 2

Hans Havermann, Nov 18 2012

Trivially all one-digit matrices are solutions, which is why 'greater than n' is specified. Two-digit matrices can never be a solution, so entries are actually greater than n^2. Most terms are three-digit solutions (less than n^3). Known exceptions are 15 digits (base 2), 7 digits (base 3), and 4 digits (bases 6, 798, 1182).

Up to base 1200, coincident terms are 28, 684, 3843, 8190, 47664, 80199, 351819, 323505, 5879259, 601524, 17159660, 20777715, respectively for base pairs (4,5), (22,25), (40,43), (81,86), (94,97), (112,115), (184,187), (276,386), (472,475), (738,749), (1061,1066), (1131,1136).

			In A219325 (base 2), the smallest number greater than 2 is 17298.
In A219324 (base 10), the smallest number greater than 10 is 247.

Hans Havermann, Table of n, a(n) for n = 2..1200
Hans Havermann, log plot to base 1200
Eric W. Weisstein, MathWorld: Circulant Matrix

Cf. A219324 (base 10), A219325 (base 2).

dcm[n_,b_] := (l = IntegerDigits[n,b]; Det[NestList[RotateRight, l, Length[l]-1]]); Table[i=b; While[dcm[i,b] != i, i++]; i, {b, 2, 43}]

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",")))

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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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A306595 Determinant of the circulant matrix whose first column corresponds to the binary digits of n.

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A219357 a(n) = smallest number greater than n, equal to the determinant of the circulant matrix formed by its base-n digits.

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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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