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

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 334, 65, 42, 76, 455, 41, 40, 98, 123, 667, 64, 52, 778, 788, 51, 50, 899, 63, 86, 7787, 2025885, 8788, 62, 74, 46996, 61, 60, 66898, 67997, 85, 73, 78998, 88899, 88999, 335, 72, 4579975, 878888, 71, 70, 10243, 5354, 355, 989999, 114

Offset: 0

Author

Paolo P. Lava, Feb 27 2019

Keywords

• nonn
• base

Comments

Here only the least numbers are listed: e.g., a(75) = 1031, even if 10002110 also produces 75.

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

Examples

			                        | 3 3 4 |
a(10) = 334 because det | 4 3 3 | = 10
                        | 3 4 3 |
.
and 334 is the least number to have this property.
.
                          | 4 6 9 9 6 |
                          | 6 4 6 9 9 |
a(34) = 46996 because det | 9 6 4 6 9 | = 34
                          | 9 9 6 4 6 |
                          | 6 9 9 6 4 |
.
and 46996 is the least number to have this property.
		

Links

• Paolo P. Lava, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A177894, A219324, A267623, A283508, A306662, A323485, A323486.

Programs

Maple

with(linalg): P:=proc(q) local a,b,c,d,j,k,i,n,t;
print(0); for i from 1 to q do for n from 1 to q do
d:=ilog10(n)+1; a:=convert(n, base, 10); c:=[];
for k from 1 to nops(a) do c:=[op(c), a[-k]]; od; t:=[op([]), c];
for k from 2 to d do b:=[op([]), c[nops(c)]];
for j from 1 to nops(c)-1 do
b:=[op(b), c[j]]; od;  c:=b; t:=[op(t), c]; od;
if i=det(t) then print(n); break; fi; od; od; end: P(10^7);

PARI

md(n) = my(d = if (n, digits(n), [0])); matdet(matrix(#d, #d, i, j, d[1+lift(Mod(j-i, #d))]));
a(n) = my(k=0); while(md(k) != n, k++); k; \\ Michel Marcus, Mar 20 2019

Formula

A177894(a(n)) = n when a(n) >= 0. - Rémy Sigrist, Feb 27 2019