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

Author

Vladimir Reshetnikov, Sep 08 2010

Keywords

• nonn
• base
• more

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

Examples

			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.

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)

Extensions

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