cp's OEIS Frontend

A266347 Numbers that cannot be represented as the product of two numbers with an equal number of significant digits (bits) in their binary representations.

%I A266347 #14 Jan 12 2016 09:38:11
%S A266347 2,3,5,7,8,10,11,12,13,14,15,17,18,19,21,22,23,26,27,29,31,32,33,34,
%T A266347 37,38,39,40,41,43,44,45,46,47,48,50,51,52,53,54,55,56,57,58,59,60,61,
%U A266347 62,63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,82,83,84,85,86,87,89,91,92,93,94,95,97,98,101,102,103,105
%N A266347 Numbers that cannot be represented as the product of two numbers with an equal number of significant digits (bits) in their binary representations.
%C A266347 All primes p are in the sequence since the only pair of divisors of p is {1, p} and since the smallest p = 2 has more bits than 1; all larger primes written in binary will require at least 2 bits to represent p. Thus A000040 is a subsequence of this sequence. - _Michael De Vlieger_, Dec 30 2015
%H A266347 Antti Karttunen, <a href="/A266347/b266347.txt">Table of n, a(n) for n = 1..10000</a>
%e A266347 From _Michael De Vlieger_, Dec 30 2015: (Start)
%e A266347 Consider pairs of divisors {d, d'} of n, both integers such that d * d' = n:
%e A266347 2 is a term, since the only pair of divisors of 2 written in binary are {1, 10}, with unequal numbers of bits.
%e A266347 3 is a term, since the only pair of divisors of 3 written in binary are {1, 11}, with unequal numbers of bits.
%e A266347 8 is a term, since the pair of divisors of 8 written in binary are {1, 100} and {10, 100}, both with unequal numbers of bits.
%e A266347 12 is a term, since the elements of {1, 1100}, {10, 110}, and {11, 100} are both unequal in length in all cases.
%e A266347 ...
%e A266347 (End)
%t A266347 Position[#, k_ /; k == 0] &@ Map[Length, Table[Flatten@ Map[Differences@ IntegerLength[#, 2] &, Transpose@ {#, n/#}] &@ TakeWhile[Divisors@ n, # <= Sqrt@ n &], {n, 100}] /. k_ /; k > 0 -> Nothing] // Flatten (* _Michael De Vlieger_, Dec 30 2015 *)
%Y A266347 Positions of zeros in A266342.
%Y A266347 Cf. A266346 (complement).
%Y A266347 Cf. A000040 (a subsequence).
%K A266347 nonn,base
%O A266347 1,1
%A A266347 _Antti Karttunen_, Dec 28 2015