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%I #69 Oct 18 2021 15:22:44
%S 1,2,3,4,5,6,7,8,9,247,370,378,407,481,518,592,629,1360,3075,26027,
%T 26933,45018,69781,80487,154791,1920261,2137616,2716713,3100883,
%U 3480140,3934896,4179451,4830936,5218958,11955168,80651025,95738203,257059332,278945612,456790123,469135802,493827160,494376160
%N Positive integers n that are equal to the determinant of the circulant matrix formed by the decimal digits of n.
%C 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.
%C 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
%C a(58) > 6*10^11. - _Giovanni Resta_, Dec 14 2012
%C 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
%H Giovanni Resta, <a href="/A219324/b219324.txt">Table of n, a(n) for n = 1..57</a> (first 47 terms from Robert G. Wilson v)
%H Max Alekseyev, <a href="/A219324/a219324.png">Illustration for a(40) = 456790123</a>
%H N. I. Belukhov, <a href="http://www.mccme.ru/free-books/matpros/mpf.pdf">Solution to Problem 14.7</a> (in Russian), Matematicheskoe Prosveshchenie 15 (2011), pp. 241-244.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Circulant_matrix">Circulant matrix</a>
%e | 2 4 7 |
%e 247 = det | 7 2 4 |
%e | 4 7 2 |
%t 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 *)
%t 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 *)
%o (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 }
%o (Python)
%o from sympy import Matrix
%o A219324_list = []
%o for n in range(1,10**4):
%o s = [int(d) for d in str(n)]
%o m = len(s)
%o if n == Matrix(m, m, lambda i, j: s[(i-j) % m]).det():
%o A219324_list.append(n) # _Chai Wah Wu_, Oct 18 2021
%Y Cf. A219325 (binary digits), A219326 (digits in reverse order), A219327 (absolute value of determinant), A306853 (permanent).
%Y Cf. A303260.
%K base,nonn,nice
%O 1,2
%A _Max Alekseyev_, Nov 17 2012
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