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

0,3

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).

Links

Table of n, a(n) for n=0..36.

Example

                                      | 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.

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);

Crossrefs

Cf. A083329, A219324, A306593, A323485, A323486.

Sequence in context: A175942 A181669 A362082 * A052940 A191304 A102444

Adjacent sequences: A306659 A306660 A306661 * A306663 A306664 A306665

Keyword

nonn,base

Author

Paolo P. Lava, Mar 04 2019

Extensions

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

Status

approved