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

2, 3, 5, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 26, 27, 29, 31, 32, 33, 34, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 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

Offset: 1

Author

Antti Karttunen, Dec 28 2015

Keywords

• nonn
• base

Comments

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

Examples

			From _Michael De Vlieger_, Dec 30 2015: (Start)
Consider pairs of divisors {d, d'} of n, both integers such that d * d' = n:
2 is a term, since the only pair of divisors of 2 written in binary are {1, 10}, with unequal numbers of bits.
3 is a term, since the only pair of divisors of 3 written in binary are {1, 11}, with unequal numbers of bits.
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.
12 is a term, since the elements of {1, 1100}, {10, 110}, and {11, 100} are both unequal in length in all cases.
...
(End)
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Positions of zeros in A266342.

Cf. A266346 (complement).

Cf. A000040 (a subsequence).

Programs

Mathematica

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 *)