A265182 -id:A265182 - cp's OEIS frontend

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

Marko Riedel, Nov 29 2015

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.

			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.

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.

Allowing A,B,C to be zero gives A265236.
See A265182, A265183 for decimal analogs.
Cf. A165416.

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

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

A265183 Base-10 analog of Marko Riedel's A265008, but allowing A, B, C to be zero.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 8, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 0, 0, 1, 1, 0, 0, 0, 0, 5, 5, 0, 0, 0, 0, 1, 0, 0, 1, 5, 5, 1, 0, 0, 0, 0, 0, 0, 0, 5, 5, 0, 0, 0, 0, 0, 0, 0, 0, 5, 5, 0, 0, 0, 0, 0, 0, 0, 0, 5, 5, 0, 0, 0, 0, 0, 0, 0, 0, 5, 5, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Dec 04 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A265008, A265182, A265236.

Cf. A218978.

a265183 n = length [() | let cs = a218978_row n, a <- cs, b <- cs, c <- cs,
                         a * b == c || c == 0 && a * b == 0]
-- Reinhard Zumkeller, Dec 05 2015

Data corrected by Reinhard Zumkeller, Dec 05 2015

cp's OEIS Frontend

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.

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