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%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
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