A333175 If n = Product (p_j^k_j) then a(n) = Sum (a(n/p_j^k_j)), with a(1) = 1.

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 6, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 6, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 6, 1, 2, 2, 1, 2, 6, 1, 2, 2, 6, 1, 2, 1, 2, 2, 2, 2, 6, 1, 2, 1, 2, 1, 6, 2, 2, 2, 2, 1, 6, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2

Offset: 1

Author

Ilya Gutkovskiy, Mar 11 2020

Keywords

• nonn

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

Examples

			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)
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Wikipedia, Permutation pattern
• Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Crossrefs

Cf. A000142, A000961 (positions of 1's), A001221, A050363, A066504, A069513, A064372, A093320, A292586.

Dominates A335451.

Permutations of prime indices are A008480.

Unsorted prime signature is A124010. Sorted prime signature is A118914.

(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.

Cf. A056239, A112798, A181796, A333221, A335452, A335463, A335521.

Programs

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}]

Formula

a(1) = 1; a(n) = Sum_{d|n, d < n, gcd(d, n/d) = 1} A069513(n/d) * a(d).

a(n) = A000142(A001221(n)).