OFFSET
1,12
COMMENTS
Depends only on sorted prime signature (A118914).
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).
LINKS
FORMULA
EXAMPLE
The a(n) permutations for n = 4, 12, 24, 48, 36, 72, 60:
(11) (112) (1112) (11112) (1122) (11122) (1123)
(121) (1121) (11121) (1212) (11212) (1132)
(211) (1211) (11211) (1221) (11221) (1213)
(2111) (12111) (2112) (12112) (1231)
(21111) (2121) (12121) (1312)
(2211) (12211) (1321)
(21112) (2113)
(21121) (2131)
(21211) (2311)
(22111) (3112)
(3121)
(3211)
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[Select[Permutations[primeMS[n]], !UnsameQ@@#&]], {n, 100}]
CROSSREFS
Positions of zeros are A005117 (squarefree numbers).
The case where the match must be contiguous is A333175.
The avoiding version is A335489.
The (1,1,1)-matching case is A335510.
Patterns are counted by A000670.
Permutations of prime indices are counted by A008480.
(1,1)-matching patterns are counted by A019472.
(1,1)-matching compositions are counted by A261982.
STC-numbers of permutations of prime indices are A333221.
Patterns matched by standard compositions are counted by A335454.
Dimensions of downsets of standard compositions are A335465.
(1,1)-matching compositions are ranked by A335488.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 14 2020
STATUS
approved