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
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
LINKS
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
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
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
CROSSREFS
KEYWORD
base,nonn,nice
AUTHOR
Max Alekseyev, Nov 17 2012
STATUS
approved