A266347 -id:A266347 - cp's OEIS frontend

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

Antti Karttunen, Dec 27 2015

			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.)

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

Cf. A000523.
Cf. A266346 (positions of nonzeros), A266347 (positions of zeros).
Cf. A266343 (positions of records).
Cf. also A266344.

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 *)

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)));

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.)

A266346 Numbers that can be represented as a product of two numbers with an equal number of significant digits (bits) in binary system.

Original entry on oeis.org

0, 1, 4, 6, 9, 16, 20, 24, 25, 28, 30, 35, 36, 42, 49, 64, 72, 80, 81, 88, 90, 96, 99, 100, 104, 108, 110, 112, 117, 120, 121, 126, 130, 132, 135, 140, 143, 144, 150, 154, 156, 165, 168, 169, 180, 182, 195, 196, 210, 225, 256, 272, 288, 289, 304, 306, 320, 323, 324, 336, 340, 342, 352, 357, 360, 361, 368, 374, 378, 380, 384, 391

Offset: 0

Antti Karttunen, Dec 28 2015

Indexing starts from zero as a(0) = 0 is a special case in this sequence.

			1 can be represented as 1*1 (1 being "1" also in base-2 system), thus it is included.
4 can be represented as 2*2, and like any square, is included.
6 can be represented as 2*3, and both "10" and "11" require two bits in binary system, thus 6 is included.

Antti Karttunen, Table of n, a(n) for n = 0..9616

Positions of nonzeros in A266342.
Cf. A266347 (complement).
Cf. A000290, A085721, A261073, A261074, A261075 (subsequences).
Cf. also A266342.

{0}~Join~Flatten[Position[#, k_ /; k > 0] &@ Table[Length@ DeleteCases[Flatten@ Map[Differences@ IntegerLength[#, 2] &, Transpose@ {#, n/#}] &@ TakeWhile[Divisors@ n, # <= Sqrt@ n &], k_ /; k > 0], {n, 400}]] (* Michael De Vlieger, Dec 30 2015 *)

cp's OEIS Frontend

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).

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