A306662 Least number k such that the determinant of the circulant matrix of its representation in base 2 is equal to n.

0, 1, 5, 11, 23, 47, 95, 191, 43, 38, 1535, 3071, 571, 12287, 24575, 137, 269, 196607, 393215, 786431, 295, 687, 6291455, 12582911, 69, 155, 100663295, 134, 293, 805306367, 1610612735, 3221225471, 75, 518, 25769803775, 301, 8874

Offset: 0

Author

Paolo P. Lava, Mar 04 2019

Keywords

• nonn
• base

Comments

Here only the least numbers are listed: e.g., a(10) = 1531, even if 1791, 1919, 1983, 2015, 2031, 2039, 2043, etc. also produce 10.

The sequence is infinite because any number of the form 3*2^(n-1) - 1 (A083329) has the determinant of the circulant matrix of its representation in base 2 equal to n but, in general, it is not the least possible term.

It would be nice to characterize the values of n where k < A083329(n).

Examples

			                                      | 1 0 1 1 |
a(3) = 11 because 11 = 1011_2 and det | 1 1 0 1 | = 3
                                      | 1 1 1 0 |
                                      | 0 1 1 1 |
.
and 11 is the least number to have this property.
.
                                       | 1 0 1 1 1 |
                                       | 1 1 0 1 1 |
a(4) = 23 because 23 = 10111_2 and det | 1 1 1 0 1 | = 4
                                       | 1 1 1 1 0 |
                                       | 0 1 1 1 1 |
.
and 23 is the least number to have this property.
		

Crossrefs

Cf. A083329, A219324, A306593, 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
a:=convert(n, base, 2); d:=nops(a); 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);

Extensions

a(31)-a(36) from Giovanni Resta, Mar 05 2019