A320386 a(n) is the smallest positive integer such that the binary representation of n*a(n) is a "binary square" (i.e., a term of A020330).

3, 5, 1, 9, 2, 6, 9, 17, 4, 1, 17, 3, 17, 17, 1, 33, 8, 2, 33, 33, 3, 24, 33, 22, 33, 33, 2, 33, 33, 22, 33, 65, 16, 4, 65, 1, 65, 65, 22, 52, 65, 22, 65, 12, 1, 65, 65, 11, 65, 52, 3, 40, 65, 1, 12, 65, 11, 65, 65, 11, 65, 65, 1, 129, 32, 8, 129, 2, 11, 39

Offset: 1

Author

Jeffrey Shallit, Oct 12 2018

Keywords

• nonn
• base

Comments

a(n) exists because if n has t bits, then (2^t+1)*n is a binary square.

Examples

			a(5) = 2 because 5 is not a binary square, but 5*2 = 10 is (its binary representation is 1010).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..8191

Crossrefs

Cf. A020330.

Programs

Maple

a:= proc(n) local k; for k while not (s-> (l->
      l::even and s[1..l/2]=s[l/2+1..l])(length(s)))(
      convert(convert(k*n, binary), string)) do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 12 2018

PARI

is(n) = my(L=#binary(n)\2); n>>L==bitand(n,2^L-1); \\ A020330
a(n) = my(k=1); while (!is(k*n), k++); k; \\ Michel Marcus, Oct 12 2018