A059981 Order of compositeness for the n-th composite number.

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 4, 1, 3, 1, 2, 4, 2, 1, 3, 4, 5, 2, 4, 1, 2, 1, 3, 5, 3, 2, 4, 1, 5, 6, 3, 1, 5, 1, 2, 3, 2, 1, 4, 6, 4, 3, 5, 1, 2, 6, 7, 4, 2, 1, 6, 1, 2, 3, 4, 3, 2, 1, 5, 7, 5, 1, 4, 1, 6, 2, 3, 7, 8, 1, 5, 3, 2, 1, 7, 2, 3, 4

Offset: 1

Author

Robert G. Wilson v, Mar 06 2001

Keywords

• easy
• nonn

Comments

Let c(k) = k-th composite number, let S(c) = S(c(k)) = k, the subscript of c; a(n) = order of compositeness of c(n) = 1+m where m is largest number such that S(S(..S(c(n))...)) with m S's is a composite.

Number of steps in the composite index chain for the n-th composite. - Daniel Forgues, Sep 28 2012

Examples

			16 is 9th composite number, so S(16)=9, 9 is 4th composite, so S(S(16))=4, 4 is first composite number, so S(S(S(16)))=1, not a composite number. Thus a(9)=3.
4 is the first composite number, so S(4)=1, not a composite number. Thus a(1)=1.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A049076, A022449 (composites with compositeness 1).

Programs

Mathematica

Composite[ n_Integer ] := (k = n + PrimePi[ n ] + 1; While[ k != n + PrimePi[ k ] + 1, k++ ]; k); CompositePi[ n_Integer ] := (n - 1 - PrimePi[ n ]); Attributes[ Composite ] = Attributes[ CompositePi ] = Listable; Table[ c = 1; k = CompositePi[ Composite[ n ] ]; While[ ! (PrimeQ[ k ] || k == 1), k = CompositePi[ k ]; c++ ]; c, {n, 100} ]