

A219324


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


14



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
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OFFSET

1,2


COMMENTS

Belukhov proved that if d is an odd divisor of p1, then for integers q=(p^d1)/((p1)*d) and t such that (p1)*(d1)/2 < t < (p1)*(d+1)/2 and gcd(t,d)=1, the number q*t equals the determinant of the circulant matrix formed by its basep 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. 241244.
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 *)


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^((i1)*j)))==n }


CROSSREFS

Cf. A219325 (binary digits), A219326 (digits in reverse order), A219327 (absolute value of determinant).
Sequence in context: A137667 A117954 A029966 * A085134 A229761 A004882
Adjacent sequences: A219321 A219322 A219323 * A219325 A219326 A219327


KEYWORD

base,nonn,nice


AUTHOR

Max Alekseyev, Nov 17 2012


STATUS

approved



