A362022 - OEIS

%I #17 Apr 07 2023 10:51:31

%S 3,5,7,9,11,11,15,17,15,21,23,19,27,23,23,33,35,27,39,37,31,43,47,35,

%T 45,45,39,39,59,43,63,65,47,69,47,51,75,77,55,69,83,55,87,75,63,87,95,

%U 67,63,85,71,77,107,75,91,71,79,93,119,79,123,95,79,129,93

%N a(n) is the least positive integer whose binary expansion is the concatenation of the binary expansions of two numbers whose product is n.

%C For any prime number p, a(p) is the least of the binary concatenation of p with 1 or the binary concatenation of 1 with p.

%H Michael De Vlieger, <a href="/A362022/b362022.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) <= 2*n + 1.

%F a(n) <= 2^A070939(n) + n.

%F a(n) = Min_{d | n} A163621(n/d, d).

%e The first terms, alongside their binary expansion split into two parts, are:

%e   n   a(n)  bin(a(n))

%e   --  ----  ---------

%e    1     3  1|1

%e    2     5  10|1

%e    3     7  11|1

%e    4     9  100|1

%e    5    11  101|1

%e    6    11  10|11

%e    7    15  111|1

%e    8    17  1000|1

%e    9    15  11|11

%e   10    21  1010|1

%e   11    23  1011|1

%e   12    19  100|11

%e   13    27  1101|1

%e   14    23  10|111

%e   15    23  101|11

%t Table[Min@ Map[FromDigits[Join @@ #, 2] &, Join @@ {#, Reverse /@ #}] &@ Map[IntegerDigits[#, 2] &, Transpose@{#, n/#}, {2}] &@ TakeWhile[Divisors[n], # <= Sqrt[n] &], {n, 60}] (* _Michael De Vlieger_, Apr 07 2023 *)

%o (PARI) a(n, base = 2) = { my (v = oo); fordiv (n, d, v = min(v, n/d * base^#digits(d, base) + d);); return (v); }

%o (Python)

%o from sympy import divisors

%o def a(n): return min(d+((n//d)<<d.bit_length()) for d in divisors(n))

%o print([a(n) for n in range(1, 66)]) # _Michael S. Branicky_, Apr 05 2023

%Y Cf. A070939, A163621, A362023 (decimal variant).

%K nonn,base

%O 1,1

%A _Rémy Sigrist_, Apr 05 2023