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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.
2
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
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
KEYWORD
nonn,base
AUTHOR
Rémy Sigrist, Apr 05 2023
STATUS
approved