A219324 -id:A219324 - cp's OEIS frontend

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.

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

A219325 Positive integers n that are equal to the determinant of the circulant matrix formed by the binary digits of n.

Original entry on oeis.org

1, 17298, 25947, 100990, 106090, 2718340, 36680364, 34505916416

Offset: 1

Max Alekseyev, Nov 17 2012

			Binary digits of 17298 are [1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0]. The 15 X 15 circulant matrix formed by their circular right rotations has determinant equal to 17298.

Giovanni Resta, Illustration of a(8)

Cf. A219324 (decimal version) provides references and more details.

dcmQ[n_]:=Module[{idn2=IntegerDigits[n,2]},Det[Table[RotateRight[idn2,k],{k,Length[ idn2]}]] == n]; Select[Range[3*10^6],dcmQ] (* This program generates the first 6 terms of the sequence. To generate a(7) and a(8), increase the Range constant to 3451*10^7, but the program will take a long time to run. *) (* Harvey P. Dale, Jul 30 2019 *)

a(7) from Hans Havermann and Emmanuel Vantieghem, Nov 19 2012

a(8) from Giovanni Resta, Dec 14 2012

A219326 Positive integers m that are equal to the determinant of the circulant matrix formed by the decimal digits of m in reverse order.

Original entry on oeis.org

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

Offset: 1

Max Alekseyev, Nov 17 2012

Terms with odd number of digits are the same as in A219324.

			           | 7 4 5 1 |
1547 = det | 1 7 4 5 |
           | 5 1 7 4 |
           | 4 5 1 7 |

Max Alekseyev, Table of n, a(n) for n = 1..53 (complete up to 10^10)

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

from sympy import Matrix
A219326_list = []
for n in range(1,10**6):
    s = [int(d) for d in str(n)][::-1]
    m = len(s)
    if n == Matrix(m, m, lambda i, j: s[(i-j) % m]).det():
        A219326_list.append(n) # Chai Wah Wu, Oct 18 2021

A303260 Determinant of n X n matrix A[i,j] = (j - i - 1 mod n) + [i=j], i.e., the circulant having (n, 0, 1, ..., n-2) as first row.

Original entry on oeis.org

1, 1, 4, 28, 273, 3421, 52288, 941578, 19505545, 456790123, 11931215316, 343871642632, 10840081272265, 371026432467913, 13702802011918048, 543154131059225686, 23000016472483168305, 1036227971225610466711, 49492629462587441963140, 2497992686980609418282548, 132849300060919364474261281

Offset: 0

M. F. Hasler, Apr 23 2018

nonn

It is remarkable that for odd n, this determinant has its base n+1 digits equal to the middle row: e.g., a(9) = 456790123 is the determinant of the circulant matrix having [4,5,6,7,9,0,1,2,3] as middle row.

a(0) = 1 is (by convention) the determinant of a 0 X 0 matrix.

			a(5) = 3421 is the determinant of the matrix
   ( 5 0 1 2 3 )
   ( 3 5 0 1 2 )
   ( 2 3 5 0 1 )  and 3421 = 23501[6], i.e., written in base 6.
   ( 1 2 3 5 0 )
   ( 0 1 2 3 5 ).

Max Alekseyev, Illustration for a(9) = 456790123 = A219324(20).
N. I. Belukhov, Solution to Problem 14.7 (in Russian), Matematicheskoe Prosveshchenie 15 (2011), pp. 241-244.
Wikipedia, Circulant matrix.

Cf. A081131(n+1) = determinant of the circulant matrix C(n) defined in formula, A070896 (signed variant).

A306593 Least number k such that the determinant of the circulant matrix formed by its decimal digits is equal to n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 334, 65, 42, 76, 455, 41, 40, 98, 123, 667, 64, 52, 778, 788, 51, 50, 899, 63, 86, 7787, 2025885, 8788, 62, 74, 46996, 61, 60, 66898, 67997, 85, 73, 78998, 88899, 88999, 335, 72, 4579975, 878888, 71, 70, 10243, 5354, 355, 989999, 114

Offset: 0

Paolo P. Lava, Feb 27 2019

Here only the least numbers are listed: e.g., a(75) = 1031, even if 10002110 also produces 75.

The sequence is infinite because any number of the form (91*10^n - 10) / 90 for n > 0 (A267623 or A283508) has the determinant of the circulant matrix equal to n but, in general, it is not the least possible term. - Giovanni Resta, Mar 06 2019

			                        | 3 3 4 |
a(10) = 334 because det | 4 3 3 | = 10
                        | 3 4 3 |
.
and 334 is the least number to have this property.
.
                          | 4 6 9 9 6 |
                          | 6 4 6 9 9 |
a(34) = 46996 because det | 9 6 4 6 9 | = 34
                          | 9 9 6 4 6 |
                          | 6 9 9 6 4 |
.
and 46996 is the least number to have this property.

Paolo P. Lava, Table of n, a(n) for n = 0..100

Cf. A177894, A219324, A267623, A283508, A306662, A323485, A323486.

with(linalg): P:=proc(q) local a,b,c,d,j,k,i,n,t;
print(0); for i from 1 to q do for n from 1 to q do
d:=ilog10(n)+1; a:=convert(n, base, 10); c:=[];
for k from 1 to nops(a) do c:=[op(c), a[-k]]; od; t:=[op([]), c];
for k from 2 to d do b:=[op([]), c[nops(c)]];
for j from 1 to nops(c)-1 do
b:=[op(b), c[j]]; od;  c:=b; t:=[op(t), c]; od;
if i=det(t) then print(n); break; fi; od; od; end: P(10^7);

md(n) = my(d = if (n, digits(n), [0])); matdet(matrix(#d, #d, i, j, d[1+lift(Mod(j-i, #d))]));
a(n) = my(k=0); while(md(k) != n, k++); k; \\ Michel Marcus, Mar 20 2019

A177894(a(n)) = n when a(n) >= 0. - Rémy Sigrist, Feb 27 2019

A323485 Least number k such that the determinant of the circulant matrix formed by its decimal digits is equal to k/n.

Original entry on oeis.org

1, 50, 648, 364, 20, 54, 21, 5000, 243, 10, 1636448, 324, 63414, 756, 73170, 432, 20043, 39366, 2121426, 46500, 6549795, 16236, 8490312, 303264, 200, 60450, 426465, 112, 27347, 2510460, 4464, 23616, 24354, 9282, 4253865, 3012552, 94017, 14022, 21411, 41000

Offset: 1

Paolo P. Lava, Jan 17 2019

a(10^j) = 10^j, with j >= 0.

			      det | 1 | = 1 = 1/1.
.
      det | 5 0 | = 25 = 50/2.
          | 0 5 |
.
          | 6 4 8 |
      det | 8 6 4 | = 216 = 648/3.
          | 4 8 6 |

Eric Weisstein's World of Mathematics, Circulant Matrix.
Wikipedia, Circulant matrix.

Cf. A219324, A323486.

with(linalg): P:=proc(q) local a,b,c,d,i,j,k,n,t;
for i from 1 to q do for n from 1 to q do
d:=ilog10(n)+1; a:=convert(n,base,10); c:=[];
for k from 1 to nops(a) do c:=[op(c),a[-k]]; od; t:=[op([]),c];
for k from 2 to d do b:=[op([]),c[nops(c)]]; for j from 1 to nops(c)-1 do
b:=[op(b),c[j]]; od;  c:=b; t:=[op(t),c]; od; if n=i*det(t) then
print(n); break; fi; od; od; end: P(10^7);

A323486 Least number k such that the determinant of the circulant matrix formed by its decimal digits is equal to n*k.

Original entry on oeis.org

1, 10168, 119700, 196, 1973082, 63980523693, 167037139360, 1350720096, 1543479071, 17239680, 4000206089, 219566358180, 104171259465, 2380649994, 113323907385, 14059155927, 19925280

Offset: 1

Paolo P. Lava, Jan 17 2019

			      det | 1 | = 1 = 1*1.
.
          | 1 0 1 6 8|
          | 8 1 0 1 6|
      det | 6 8 1 0 1| = 20336 = 2*10168.
          | 1 6 8 1 0|
          | 0 1 6 8 1|

Eric Weisstein's World of Mathematics, Circulant Matrix.
Wikipedia, Circulant matrix.

Cf. A219324, A323485.

with(linalg): P:=proc(q) local a,b,c,d,i,j,k,n,t;
for i from 1 to q do for n from 1 to q do
d:=ilog10(n)+1; a:=convert(n,base,10); c:=[];
for k from 1 to nops(a) do c:=[op(c),a[-k]]; od; t:=[op([]),c];
for k from 2 to d do b:=[op([]),c[nops(c)]]; for j from 1 to nops(c)-1 do
b:=[op(b),c[j]]; od;  c:=b; t:=[op(t),c]; od; if i*n=det(t) then
print(n); break; fi; od; od; end: P(10^7);

a(6)-a(17) from Giovanni Resta, Jan 21 2019

A306662 Least number k such that the determinant of the circulant matrix of its representation in base 2 is equal to n.

Original entry on oeis.org

0, 1, 5, 11, 23, 47, 95, 191, 43, 38, 1535, 3071, 571, 12287, 24575, 137, 269, 196607, 393215, 786431, 295, 687, 6291455, 12582911, 69, 155, 100663295, 134, 293, 805306367, 1610612735, 3221225471, 75, 518, 25769803775, 301, 8874

Offset: 0

Paolo P. Lava, Mar 04 2019

Here only the least numbers are listed: e.g., a(10) = 1531, even if 1791, 1919, 1983, 2015, 2031, 2039, 2043, etc. also produce 10.
The sequence is infinite because any number of the form 3*2^(n-1) - 1 (A083329) has the determinant of the circulant matrix of its representation in base 2 equal to n but, in general, it is not the least possible term.
It would be nice to characterize the values of n where k < A083329(n).

			                                      | 1 0 1 1 |
a(3) = 11 because 11 = 1011_2 and det | 1 1 0 1 | = 3
                                      | 1 1 1 0 |
                                      | 0 1 1 1 |
.
and 11 is the least number to have this property.
.
                                       | 1 0 1 1 1 |
                                       | 1 1 0 1 1 |
a(4) = 23 because 23 = 10111_2 and det | 1 1 1 0 1 | = 4
                                       | 1 1 1 1 0 |
                                       | 0 1 1 1 1 |
.
and 23 is the least number to have this property.

Cf. A083329, A219324, A306593, A323485, A323486.

with(linalg): P:=proc(q) local a, b, c, d, j, k, i, n, t;
print(0); for i from 1 to q do for n from 1 to q do
a:=convert(n, base, 2); d:=nops(a); c:=[];
for k from 1 to nops(a) do c:=[op(c), a[-k]]; od; t:=[op([]), c];
for k from 2 to d do b:=[op([]), c[nops(c)]];
for j from 1 to nops(c)-1 do
b:=[op(b), c[j]]; od;  c:=b; t:=[op(t), c]; od;
if i=det(t) then print(n); break; fi; od; od; end: P(10^7);

a(31)-a(36) from Giovanni Resta, Mar 05 2019

A306853 Positive integers equal to the permanent of the circulant matrix formed by their decimal digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 261, 370, 407, 52036, 724212, 223123410

Offset: 1

Paolo P. Lava, Mar 13 2019

1, 2, 3, 4, 5, 6, 7, 8, 9, 370 and 407 are also equal to the determinant of the circulant matrix formed by their decimal digits.

			     | 2 6 1 |
perm | 1 2 6 | = 2*2*2 + 6*6*6 + 1*1*1 + 1*2*6 + 6*1*2 + 2*6*1 = 261.
     | 6 1 2 |
.
     | 2 2 3 1 2 3 4 1 0 |
     | 0 2 2 3 1 2 3 4 1 |
     | 1 0 2 2 3 1 2 3 4 |
     | 4 1 0 2 2 3 1 2 3 |
perm | 3 4 1 0 2 2 3 1 2 | = 223123410
     | 2 3 4 1 0 2 2 3 1 |
     | 1 2 3 4 1 0 2 2 3 |
     | 3 1 2 3 4 1 0 2 2 |
     | 2 3 1 2 3 4 1 0 2 |

Eric Weisstein's World of Mathematics, Permanent
Eric Weisstein's World of Mathematics, Circulant Matrix

Cf. A219324, A219327, A306662, A306593, A306714.

Up to n=110 the permanent of the circulant matrix of the digits of n is equal to A101337 but from n=111 on it can differ.

with(linalg): P:=proc(q) local a, b, c, d, i, j, k, n, t;
for n from 1 to q do d:=ilog10(n)+1; a:=convert(n, base, 10); c:=[];
for k from 1 to nops(a) do c:=[op(c), a[-k]]; od; t:=[op([]), c];
for k from 2 to d do b:=[op([]), c[nops(c)]];
for j from 1 to nops(c)-1 do b:=[op(b), c[j]]; od;
c:=b; t:=[op(t), c]; od; if n=permanent(t)
then print(n); fi; od; end: P(10^7);

mpd(n) = {my(d = digits(n)); matpermanent(matrix(#d, #d, i, j, d[1+lift(Mod(j-i, #d))]));}
isok(n) = mpd(n) == n; \\ Michel Marcus, Mar 14 2019

from sympy import Matrix
A306853_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]).per():
        A306853_list.append(n) # Chai Wah Wu, Oct 18 2021

a(15) from Vaclav Kotesovec, Aug 19 2021

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}]

cp's OEIS Frontend

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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A219325 Positive integers n that are equal to the determinant of the circulant matrix formed by the binary digits of n.

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A219326 Positive integers m that are equal to the determinant of the circulant matrix formed by the decimal digits of m in reverse order.

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A303260 Determinant of n X n matrix A[i,j] = (j - i - 1 mod n) + [i=j], i.e., the circulant having (n, 0, 1, ..., n-2) as first row.

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A306593 Least number k such that the determinant of the circulant matrix formed by its decimal digits is equal to n.

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A323485 Least number k such that the determinant of the circulant matrix formed by its decimal digits is equal to k/n.

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A323486 Least number k such that the determinant of the circulant matrix formed by its decimal digits is equal to n*k.

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A306662 Least number k such that the determinant of the circulant matrix of its representation in base 2 is equal to n.

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