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

1,1

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.

Links

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

Formula

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

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

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

Example

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

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 *)

Prog

(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)<<d.bit_length()) for d in divisors(n))
print([a(n) for n in range(1, 66)]) # Michael S. Branicky, Apr 05 2023

Crossrefs

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

Sequence in context: A206544 A122799 A345425 * A162495 A107315 A340855

Adjacent sequences: A362019 A362020 A362021 * A362023 A362024 A362025

Keyword

nonn,base

Author

Rémy Sigrist, Apr 05 2023

Status

approved