OFFSET
1,6
COMMENTS
Number of ordered prime factorizations of radical of n.
Number of permutations of the prime indices of n (counting multiplicity) avoiding the patterns (1,2,1) and (2,1,2). These are permutations with all equal parts contiguous. Depends only on sorted prime signature (A118914). - Gus Wiseman, Jun 27 2020
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Permutation pattern
FORMULA
EXAMPLE
From Gus Wiseman, Jun 27 2020 (Start)
The a(n) permutations of prime indices for n = 2, 12, 60:
(1) (112) (1123)
(211) (1132)
(2113)
(2311)
(3112)
(3211)
(End)
MAPLE
f:= n -> nops(numtheory:-factorset(n))!:
map(f, [$1..100]); # Robert Israel, Mar 12 2020
MATHEMATICA
a[1] = 1; a[n_] := a[n] = Plus @@ (a[n/#[[1]]^#[[2]]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 100}]
a[1] = 1; a[n_] := a[n] = Sum[If[GCD[n/d, d] == 1 && d < n, Boole[PrimePowerQ[n/d]] a[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 100}]
Table[PrimeNu[n]!, {n, 1, 100}]
CROSSREFS
Cf. A000142, A000961 (positions of 1's), A001221, A050363, A066504, A069513, A064372, A093320, A292586.
Dominates A335451.
Permutations of prime indices are A008480.
(1,2,1)-avoiding permutations of prime indices are A335449.
(2,1,2)-avoiding permutations of prime indices are A335450.
(1,2,1) or (2,1,2)-matching permutations of prime indices are A335460.
(1,2,1) and (2,1,2)-matching permutations of prime indices are A335462.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Mar 11 2020
STATUS
approved