A298971 Number of compositions of n that are proper powers of Lyndon words.

0, 1, 1, 2, 1, 4, 1, 5, 3, 8, 1, 16, 1, 20, 9, 35, 1, 69, 1, 110, 21, 188, 1, 381, 7, 632, 59, 1184, 1, 2300, 1, 4115, 189, 7712, 25, 14939, 1, 27596, 633, 52517, 1, 101050, 1, 190748, 2247, 364724, 1, 703331, 19, 1342283, 7713, 2581430, 1, 4985609, 193

Offset: 1

Author

Gus Wiseman, Jan 30 2018

Keywords

• nonn

Comments

a(n) is the number of compositions of n that are not Lyndon words but are of the form p * p * ... * p where * is concatenation and p is a Lyndon word.

Examples

			The a(12) = 16 compositions: 111111111111, 1111211112, 11131113, 112112112, 11221122, 114114, 12121212, 123123, 131313, 132132, 1515, 222222, 2424, 3333, 444, 66.
		

Crossrefs

Cf. A000005, A000031, A000740, A000961, A001045, A008965, A019536, A034691, A051953, A052823, A059966, A060223, A178472, A185700, A296302, A296373.

Programs

Mathematica

Table[Sum[DivisorSum[d,MoebiusMu[d/#]*(2^#-1)&]/d,{d,Most@Divisors[n]}],{n,100}]

PARI

a(n) = sumdiv(n, d, (2^d-1)*(eulerphi(n/d)-moebius(n/d))/n); \\ Michel Marcus, Jan 31 2018

Formula

a(n) = Sum_{d|n} (2^d-1)*(phi(n/d)-mu(n/d))/n.

a(n) = A008965(n) - A059966(n).