A219324
Positive integers n that are equal to the determinant of the circulant matrix formed by the decimal digits of n.
Original entry on oeis.org
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
| 2 4 7 |
247 = det | 7 2 4 |
| 4 7 2 |
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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 *)
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{ 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 }
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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
A303367
Numbers equal to the determinant of a circulant matrix based on the base-7 digits of n.
Original entry on oeis.org
1, 2, 3, 4, 5, 6, 27, 81, 133, 143, 144, 152, 190, 209, 250, 1125, 8569, 10934, 16401, 237568, 362306, 391257, 695751, 723260, 5294625, 6056974, 6332291, 6523631, 6669475, 11128547, 12486285, 17417491, 18682225, 19429514, 19781014, 20924092, 21671381
Offset: 1
133 is in the sequence because 133 = 250[7] (in base 7) and 133 = det [2,5,0; 0,2,5; 5,0,2].
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(c(v)=abs(matdet(matrix(#v,#v,i,j,v[(j-i)%#v+1]))));for(n=1,oo,n==c(digits(n,7))&&print1(n","))
A303369
Numbers equal to the determinant of a circulant matrix based on the base-9 digits of n.
Original entry on oeis.org
1, 2, 3, 4, 5, 6, 7, 8, 104, 126, 130, 468, 2035, 8052, 8421, 12100, 18788, 21296, 60736, 90155, 109135, 119795, 123201, 137605, 147095, 160965, 199728, 457856, 523809, 697334, 2958176, 3652360, 4725030, 5331625, 6825024, 7014400, 7694336, 9376133, 14012352
Offset: 1
104 is in the sequence because 104 = 125[9] (in base 9) and 104 = det [1,2,5; 5,1,2; 2,5,1].
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(c(v)=abs(matdet(matrix(#v,#v,i,j,v[(j-i)%#v+1]))));for(n=1,oo,n==c(digits(n,9))&&print1(n","))
A303368
Numbers equal to the determinant of a circulant matrix based on the base-8 digits of n.
Original entry on oeis.org
1, 2, 3, 4, 5, 6, 7, 21, 133, 432, 525, 1200, 1456, 1904, 4774, 24583, 281048, 289820, 333293, 420239, 428752, 430686, 437554, 500380, 500888, 736600, 941578, 984377, 1027176, 1069975, 1112774, 1155573, 1662216, 1776201, 2087008, 3212235, 7928375, 8131725
Offset: 1
-
(c(v)=abs(matdet(matrix(#v,#v,i,j,v[(j-i)%#v+1]))));for(n=1,oo,n==c(digits(n,8))&&print1(n","))
A303366
Numbers equal to the determinant of a circulant matrix based on the base-6 digits of n.
Original entry on oeis.org
1, 2, 3, 4, 5, 320, 1936, 2761, 3421, 3732, 4043, 4354, 1697296, 2121620, 2970268, 3076349, 5222639, 8979475, 41657517, 63095140, 110416495, 488553143, 646839439, 1066194997, 1209597525, 1803581465, 1861934400, 2019744271, 2691271467, 2842031031, 3052726590, 7100897730, 10806417998
Offset: 1
320 is in the sequence because 320 = 1252[6] (in base 6) and 320 = det [1,2,5,2; 2,1,2,5; 5,2,1,2; 2,5,2,1].
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Select[Range[5000],#==Det[NestList[RotateRight,IntegerDigits[#,6],IntegerLength[#,6]-1]]&] (* The program generates the first 12 terms of the sequence. *) (* Harvey P. Dale, Mar 11 2024 *)
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(c(v)=abs(matdet(matrix(#v,#v,i,j,v[(j-i)%#v+1]))));for(n=1,oo,n==c(digits(n,6))&&print1(n","))
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
| 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 |
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.
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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);
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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
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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
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
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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
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])).
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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
⎡1 5 4 7⎤
1547 = det ⎢5 4 7 1⎥
⎢4 7 1 5⎥
⎣7 1 5 4⎦.
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Select[Range[10^6], Equal[Det[NestList[RotateLeft, #2, #3 - 1]], #1] & @@ {#1, #2, Length[#2]} & @@ {#, IntegerDigits[#]} &] (* Michael De Vlieger, Oct 18 2021 *)
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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
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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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