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A362022
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a(n) is the least positive integer whose binary expansion is the concatenation of the binary expansions of two numbers whose product is n.
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2
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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a(n) <= 2*n + 1.
a(n) = Min_{d | n} A163621(n/d, d).
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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))
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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