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

1,1

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

Links

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

Example

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)

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 *)

Crossrefs

Positions of zeros in A266342.

Cf. A266346 (complement).

Cf. A000040 (a subsequence).

Sequence in context: A076385 A152604 A302293 * A028788 A197128 A034707

Adjacent sequences: A266344 A266345 A266346 * A266348 A266349 A266350

Keyword

nonn,base

Author

Antti Karttunen, Dec 28 2015

Status

approved