A306662 - OEIS

%I #21 Mar 12 2019 13:10:01

%S 0,1,5,11,23,47,95,191,43,38,1535,3071,571,12287,24575,137,269,196607,

%T 393215,786431,295,687,6291455,12582911,69,155,100663295,134,293,

%U 805306367,1610612735,3221225471,75,518,25769803775,301,8874

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

%C 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.

%C 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.

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

%e                                       | 1 0 1 1 |

%e a(3) = 11 because 11 = 1011_2 and det | 1 1 0 1 | = 3

%e                                       | 1 1 1 0 |

%e                                       | 0 1 1 1 |

%e .

%e and 11 is the least number to have this property.

%e .

%e                                        | 1 0 1 1 1 |

%e                                        | 1 1 0 1 1 |

%e a(4) = 23 because 23 = 10111_2 and det | 1 1 1 0 1 | = 4

%e                                        | 1 1 1 1 0 |

%e                                        | 0 1 1 1 1 |

%e .

%e and 23 is the least number to have this property.

%p with(linalg): P:=proc(q) local a, b, c, d, j, k, i, n, t;

%p print(0); for i from 1 to q do for n from 1 to q do

%p a:=convert(n, base, 2); d:=nops(a); c:=[];

%p for k from 1 to nops(a) do c:=[op(c), a[-k]]; od; t:=[op([]), c];

%p for k from 2 to d do b:=[op([]), c[nops(c)]];

%p for j from 1 to nops(c)-1 do

%p b:=[op(b), c[j]]; od;  c:=b; t:=[op(t), c]; od;

%p if i=det(t) then print(n); break; fi; od; od; end: P(10^7);

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

%K nonn,base

%O 0,3

%A _Paolo P. Lava_, Mar 04 2019

%E a(31)-a(36) from _Giovanni Resta_, Mar 05 2019