A265008 Number of positive solutions to the equation A x B = C, where A, B and C are made from (contiguous) substrings of the binary representation of n.

1, 3, 3, 6, 5, 9, 5, 10, 8, 9, 9, 18, 13, 15, 7, 15, 12, 12, 13, 18, 11, 17, 13, 30, 23, 21, 17, 30, 21, 21, 9, 21, 17, 16, 16, 18, 16, 21, 17, 30, 22, 15, 17, 32, 21, 25, 19, 45, 34, 33, 27, 36, 25, 25, 25, 50, 37, 33, 27, 42, 33, 27, 11, 28, 23, 21, 21, 22, 20, 24, 20, 30, 20, 24, 23, 33, 27, 29, 21, 45, 34, 30, 29, 30, 17, 29, 25, 52, 40, 33, 25, 46, 31

Offset

1,2

Comments

Here all three of A, B and C are positive; if A and B are distinct then the solutions A X B = C and B X A = C are regarded as distinct; while A x A is counted once. A substring may be used any number of times.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Marko Riedel, Maple code for the number of solutions to A x B = C using binary substrings of n.

Example

When n=6 = (110)_2 we get the substrings 1=(1)_2, 2=(10)_2, 3=(11)_2 and 6=(110)_2 with the products 1*1=1, 1*2=2, 2*1=2, 1*3=3, 3*1=3, 1*6=6, 6*1=6, 2*3=6, 3*2=6 for a total of 9. When n=8 = (1000)_2 we get the substrings 1=(1)_2, 2=(10)_2, 4=(100)_2 and 8=(1000)_2 with the products 1*1=1, 1*2=2, 2*1=2, 1*4=4, 4*1=4, 1*8=1, 8*1=1, 2*2=4, 2*4=8, 4*2=8 for a total of 10.

Maple

F:= proc(n) local L, ss;
  L:= convert(n, base, 2);
  ss:= {seq(seq(add(2^(i-j)*L[i], i=j..k), j=1..k), k=1..nops(L))} minus {0};
  numboccur(true, [seq(seq(member(a*b, ss), a=ss), b=ss)]);
end proc:
seq(F(n), n=1..100); # Robert Israel, Nov 30 2015

Prog

(Haskell)
a265008 n = length [() | let cs = a165416_row n, c <- cs,
            let as = takeWhile (<= c) cs, a <- as, b <- as, a * b == c]
-- Reinhard Zumkeller, Dec 05 2015

Crossrefs

Allowing A,B,C to be zero gives A265236.

See A265182, A265183 for decimal analogs.

Cf. A165416.

Sequence in context: A369857 A380137 A071047 * A165606 A295220 A280167

Adjacent sequences: A265005 A265006 A265007 * A265009 A265010 A265011

Keyword

nonn,base

Author

Marko Riedel, Nov 29 2015

Status

approved