A266342 a(n) = number of ways n can be expressed as a product of two natural numbers that have same number of significant digits in base-2 representation (up to the ordering of unequal factors).

1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2

Offset: 1

Author

Antti Karttunen, Dec 27 2015

Keywords

• nonn
• base

Examples

			For n=1 we have one possibility, 1*1 = 1, thus a(1) = 1.
For n=2 we have no choices, as the binary representation of 1 which is "1" is shorter than the binary representation of 2 which is "10", thus a(2) = 0 (and likewise for any prime).
For n=120 we have two choices, either 8*15 (in binary "1000" * "1111") or 10*12 ("1010" * "1100"), thus a(120) = 2. (15*8 and 8*15 are not counted separately.)
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A000523.

Cf. A266346 (positions of nonzeros), A266347 (positions of zeros).

Cf. A266343 (positions of records).

Cf. also A266344.

Programs

Mathematica

Map[Length, Table[Flatten@ Map[Differences@ IntegerLength[#, 2] &, Transpose@ {#, n/#}] &@ TakeWhile[Divisors@ n, # <= Sqrt@ n &], {n, 120}] /. k_ /; k > 0 -> Nothing] (* Michael De Vlieger, Dec 30 2015, Version 7.0 *)

PARI

A000523(n) = if(n<1,0,#binary(n) - 1);
A266342(n) = sumdiv(n, d, ((d <= (n/d)) && (A000523(d)==A000523(n/d))));
for(n=1, 32768, write("b266342.txt", n, " ", A266342(n)));

Formula

a(n) = Sum_{d|n} [(d <= (n/d)) and (A000523(d) = A000523(n/d))].

(In the above formula [ ] stands for Iverson bracket, resulting in 1 only if d is less than or equal to n/d and the binary lengths of d and n/d are equal, and 0 otherwise.)