A266347 - OEIS

%I #14 Jan 12 2016 09:38:11

%S 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 37,38,39,40,41,43,44,45,46,47,48,50,51,52,53,54,55,56,57,58,59,60,61,

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

%C 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 Antti Karttunen, <a href="/A266347/b266347.txt">Table of n, a(n) for n = 1..10000</a>

%e From _Michael De Vlieger_, Dec 30 2015: (Start)

%e Consider pairs of divisors {d, d'} of n, both integers such that d * d' = n:

%e 2 is a term, since the only pair of divisors of 2 written in binary are {1, 10}, with unequal numbers of bits.

%e 3 is a term, since the only pair of divisors of 3 written in binary are {1, 11}, with unequal numbers of bits.

%e 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 12 is a term, since the elements of {1, 1100}, {10, 110}, and {11, 100} are both unequal in length in all cases.

%e ...

%e (End)

%t 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 Positions of zeros in A266342.

%Y Cf. A266346 (complement).

%Y Cf. A000040 (a subsequence).

%K nonn,base

%O 1,1

%A _Antti Karttunen_, Dec 28 2015