A219324 Positive integers n that are equal to the determinant of the circulant matrix formed by the decimal digits of n.

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

Author

Max Alekseyev, Nov 17 2012

Keywords

• base
• nonn
• nice

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

Examples

			          | 2 4 7 |
247 = det | 7 2 4 |
          | 4 7 2 |
		

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

Crossrefs

Cf. A219325 (binary digits), A219326 (digits in reverse order), A219327 (absolute value of determinant), A306853 (permanent).

Cf. A303260.

Programs

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 *)

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