A175809 a(n) is the number of shortest common superstrings of the binary representations of all natural numbers from 1 to n.

1, 1, 1, 1, 2, 2, 2, 2, 6, 4, 6, 4, 16, 16, 16, 16, 84, 56, 120, 108, 216, 108, 1296, 972, 504, 312, 768, 448, 2048, 2048, 2048, 2048

Offset

1,5

Comments

All shortest common superstrings share the same number of ones and the same number of substrings of the form "10". If the length of the shortest common superstrings is a power of two (A175808(n) = 2^m), then we know that the lexicographically largest superstring coincides with the lexicographically largest de Bruijn sequence, B(2,m) (A166316(m)). This tells us that in this case all shortest common superstrings contain 2^(m-1) ones in 2^(m-2) groups separated by one or more zeros. - Thomas Scheuerle, Sep 19 2021

Links

Table of n, a(n) for n=1..32.

Formula

From Thomas Scheuerle, Sep 19 2021: (Start)

a(2^n) = A016031(n) (if conjectured A175808(2^n) = 2^n is true).

a(2^n-3) = a(2^n-2) for n > 2. In this case the set of superstrings is equal.

a(2^n-2) = a(2^n-1) = a(2^n) for n > 1. Conjectured. (End)

Example

a(5)=2 because there are 2 shortest common superstrings of 1,10,11,100,101; they are 110100 and 101100.

Crossrefs

Cf. A175808 (length of shortest common superstrings).

Cf. A056744 (least decimal values of shortest common superstrings).

Cf. A166316, A016031.

Sequence in context: A103222 A319100 A304794 * A061033 A075094 A238413

Adjacent sequences: A175806 A175807 A175808 * A175810 A175811 A175812

Keyword

nonn,base,more

Author

Vladimir Reshetnikov, Sep 08 2010

Extensions

a(21)-a(32) from Thomas Scheuerle, Sep 19 2021

Status

approved