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A306593 Least number k such that the determinant of the circulant matrix formed by its decimal digits is equal to n.

%I A306593 #39 Mar 21 2019 04:38:58
%S A306593 0,1,2,3,4,5,6,7,8,9,334,65,42,76,455,41,40,98,123,667,64,52,778,788,
%T A306593 51,50,899,63,86,7787,2025885,8788,62,74,46996,61,60,66898,67997,85,
%U A306593 73,78998,88899,88999,335,72,4579975,878888,71,70,10243,5354,355,989999,114
%N A306593 Least number k such that the determinant of the circulant matrix formed by its decimal digits is equal to n.
%C A306593 Here only the least numbers are listed: e.g., a(75) = 1031, even if 10002110 also produces 75.
%C A306593 The sequence is infinite because any number of the form (91*10^n - 10) / 90 for n > 0 (A267623 or A283508) has the determinant of the circulant matrix equal to n but, in general, it is not the least possible term. - _Giovanni Resta_, Mar 06 2019
%H A306593 Paolo P. Lava, <a href="/A306593/b306593.txt">Table of n, a(n) for n = 0..100</a>
%F A306593 A177894(a(n)) = n when a(n) >= 0. - _Rémy Sigrist_, Feb 27 2019
%e A306593                         | 3 3 4 |
%e A306593 a(10) = 334 because det | 4 3 3 | = 10
%e A306593                         | 3 4 3 |
%e A306593 .
%e A306593 and 334 is the least number to have this property.
%e A306593 .
%e A306593                           | 4 6 9 9 6 |
%e A306593                           | 6 4 6 9 9 |
%e A306593 a(34) = 46996 because det | 9 6 4 6 9 | = 34
%e A306593                           | 9 9 6 4 6 |
%e A306593                           | 6 9 9 6 4 |
%e A306593 .
%e A306593 and 46996 is the least number to have this property.
%p A306593 with(linalg): P:=proc(q) local a,b,c,d,j,k,i,n,t;
%p A306593 print(0); for i from 1 to q do for n from 1 to q do
%p A306593 d:=ilog10(n)+1; a:=convert(n, base, 10); c:=[];
%p A306593 for k from 1 to nops(a) do c:=[op(c), a[-k]]; od; t:=[op([]), c];
%p A306593 for k from 2 to d do b:=[op([]), c[nops(c)]];
%p A306593 for j from 1 to nops(c)-1 do
%p A306593 b:=[op(b), c[j]]; od;  c:=b; t:=[op(t), c]; od;
%p A306593 if i=det(t) then print(n); break; fi; od; od; end: P(10^7);
%o A306593 (PARI) md(n) = my(d = if (n, digits(n), [0])); matdet(matrix(#d, #d, i, j, d[1+lift(Mod(j-i, #d))]));
%o A306593 a(n) = my(k=0); while(md(k) != n, k++); k; \\ _Michel Marcus_, Mar 20 2019
%Y A306593 Cf. A177894, A219324, A267623, A283508, A306662, A323485, A323486.
%K A306593 nonn,base
%O A306593 0,3
%A A306593 _Paolo P. Lava_, Feb 27 2019