login
Number of chains of factorizations of n into factors > 1, ordered by refinement, starting with the prime factorization of n and ending with the maximum factorization (n).
5

%I #4 Jul 24 2018 21:21:36

%S 1,1,1,1,1,1,1,2,1,1,1,3,1,1,1,6,1,3,1,3,1,1,1,11,1,1,2,3,1,4,1,18,1,

%T 1,1,15,1,1,1,11,1,4,1,3,3,1,1,49,1,3,1,3,1,11,1,11,1,1,1,21,1,1,3,74,

%U 1,4,1,3,1,4,1,78,1,1,3,3,1,4,1,49,6,1,1

%N Number of chains of factorizations of n into factors > 1, ordered by refinement, starting with the prime factorization of n and ending with the maximum factorization (n).

%C If x and y are factorizations of the same integer and it is possible to produce x by further factoring the factors of y, flattening, and sorting, then x <= y.

%F a(prime^n) = A213427(n).

%e The a(24) = 11 chains:

%e (2*2*2*3) < (24)

%e (2*2*2*3) < (2*12) < (24)

%e (2*2*2*3) < (3*8) < (24)

%e (2*2*2*3) < (4*6) < (24)

%e (2*2*2*3) < (2*2*6) < (24)

%e (2*2*2*3) < (2*3*4) < (24)

%e (2*2*2*3) < (2*2*6) < (2*12) < (24)

%e (2*2*2*3) < (2*2*6) < (4*6) < (24)

%e (2*2*2*3) < (2*3*4) < (2*12) < (24)

%e (2*2*2*3) < (2*3*4) < (3*8) < (24)

%e (2*2*2*3) < (2*3*4) < (4*6) < (24)

%Y Cf. A001055, A002846, A007716, A045778, A162247, A213427, A275024, A281113, A299202, A317144, A317145, A317146.

%K nonn

%O 1,8

%A _Gus Wiseman_, Jul 23 2018