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",")))
A330519
Circulant determinant chain starting at 1007749.
Original entry on oeis.org
1007749, 18791108, -186878195, -1050017609, -9619142907, -4098030816, -59235488169, -43035664380, -25214990766, -189627605493, 779029018560, 14532890711040, 364989611366055, 1929185790836472, -1121729573175075, 13492031465869425, 329600773788765624
Offset: 1
A323548
Amicable numbers under the calculation of the determinant of the circulant matrix formed by their decimal digits.
Original entry on oeis.org
108, 182, 473, 513, 1139005, 3798233, 142250866, 186519853, 245578912, 387304234, 12410397495, 15303786345, 28309184956, 28670744905
Offset: 1
| 1 0 8 | | 5 1 3 |
det | 8 1 0 | = 513 and det | 3 5 1 | = 108.
| 0 8 1 | | 1 3 5 |
.
| 1 8 2 | | 4 7 3 |
det | 2 1 8 | = 473 and det | 3 4 7 | = 182.
| 8 2 1 | | 7 3 4 |
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with(linalg): P:=proc(q) local a,b,c,d,j,k,n,p,t,x,y:
for n from 1 to q do x:=n: for p from 1 to 2 do
d:=ilog10(x)+1: a:=convert(x,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; x:=det(t): if x=0 then break:
else if p=1 then y:=x: fi: fi: od:
if n=x and y<>x then print(n); fi: od: end: P(10^8):
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is(n) = my(c = amidet(n)); if(c == n, return(0)); amidet(c) == n
amidet(n) = my(d = digits(n), qd = #d, m = matrix(qd, qd)); for(i = 1, qd, for(j = 1, qd, m[i, j] = d[1 + (j - i)%qd])); ami = matdet(m); ami \\ David A. Corneth, Jan 21 2019
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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