A362022 a(n) is the least positive integer whose binary expansion is the concatenation of the binary expansions of two numbers whose product is n.

3, 5, 7, 9, 11, 11, 15, 17, 15, 21, 23, 19, 27, 23, 23, 33, 35, 27, 39, 37, 31, 43, 47, 35, 45, 45, 39, 39, 59, 43, 63, 65, 47, 69, 47, 51, 75, 77, 55, 69, 83, 55, 87, 75, 63, 87, 95, 67, 63, 85, 71, 77, 107, 75, 91, 71, 79, 93, 119, 79, 123, 95, 79, 129, 93

Offset: 1

Author

Rémy Sigrist, Apr 05 2023

Keywords

• nonn
• base

Comments

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.

Examples

			The first terms, alongside their binary expansion split into two parts, are:
  n   a(n)  bin(a(n))
  --  ----  ---------
   1     3  1|1
   2     5  10|1
   3     7  11|1
   4     9  100|1
   5    11  101|1
   6    11  10|11
   7    15  111|1
   8    17  1000|1
   9    15  11|11
  10    21  1010|1
  11    23  1011|1
  12    19  100|11
  13    27  1101|1
  14    23  10|111
  15    23  101|11
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Mathematica

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 *)

PARI

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

Python

from sympy import divisors
def a(n): return min(d+((n//d)<Michael S. Branicky, Apr 05 2023

Formula

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

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

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