cp's OEIS Frontend

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.

%I A362022 #17 Apr 07 2023 10:51:31
%S A362022 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 A362022 45,45,39,39,59,43,63,65,47,69,47,51,75,77,55,69,83,55,87,75,63,87,95,
%U A362022 67,63,85,71,77,107,75,91,71,79,93,119,79,123,95,79,129,93
%N 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.
%C A362022 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 A362022 Michael De Vlieger, <a href="/A362022/b362022.txt">Table of n, a(n) for n = 1..10000</a>
%F A362022 a(n) <= 2*n + 1.
%F A362022 a(n) <= 2^A070939(n) + n.
%F A362022 a(n) = Min_{d | n} A163621(n/d, d).
%e A362022 The first terms, alongside their binary expansion split into two parts, are:
%e A362022   n   a(n)  bin(a(n))
%e A362022   --  ----  ---------
%e A362022    1     3  1|1
%e A362022    2     5  10|1
%e A362022    3     7  11|1
%e A362022    4     9  100|1
%e A362022    5    11  101|1
%e A362022    6    11  10|11
%e A362022    7    15  111|1
%e A362022    8    17  1000|1
%e A362022    9    15  11|11
%e A362022   10    21  1010|1
%e A362022   11    23  1011|1
%e A362022   12    19  100|11
%e A362022   13    27  1101|1
%e A362022   14    23  10|111
%e A362022   15    23  101|11
%t A362022 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 A362022 (PARI) a(n, base = 2) = { my (v = oo); fordiv (n, d, v = min(v, n/d * base^#digits(d, base) + d);); return (v); }
%o A362022 (Python)
%o A362022 from sympy import divisors
%o A362022 def a(n): return min(d+((n//d)<<d.bit_length()) for d in divisors(n))
%o A362022 print([a(n) for n in range(1, 66)]) # _Michael S. Branicky_, Apr 05 2023
%Y A362022 Cf. A070939, A163621, A362023 (decimal variant).
%K A362022 nonn,base
%O A362022 1,1
%A A362022 _Rémy Sigrist_, Apr 05 2023