|
|
A219324
|
|
Positive integers n that are equal to the determinant of the circulant matrix formed by the decimal digits of n.
|
|
16
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
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
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
|
|
LINKS
|
|
|
EXAMPLE
|
| 2 4 7 |
247 = det | 7 2 4 |
| 4 7 2 |
|
|
MATHEMATICA
|
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 *)
|
|
PROG
|
(PARI) { 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 }
(Python)
from sympy import Matrix
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():
|
|
CROSSREFS
|
|
|
KEYWORD
|
base,nonn,nice
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|