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

%I A333175 #35 Jul 12 2020 19:48:15
%S A333175 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,
%T A333175 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,
%U A333175 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
%N A333175 If n = Product (p_j^k_j) then a(n) = Sum (a(n/p_j^k_j)), with a(1) = 1.
%C A333175 Number of ordered prime factorizations of radical of n.
%C A333175 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
%H A333175 Robert Israel, <a href="/A333175/b333175.txt">Table of n, a(n) for n = 1..10000</a>
%H A333175 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>
%H A333175 Gus Wiseman, <a href="/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>
%F A333175 a(1) = 1; a(n) = Sum_{d|n, d < n, gcd(d, n/d) = 1} A069513(n/d) * a(d).
%F A333175 a(n) = A000142(A001221(n)).
%e A333175 From _Gus Wiseman_, Jun 27 2020 (Start)
%e A333175 The a(n) permutations of prime indices for n = 2, 12, 60:
%e A333175   (1)  (112)  (1123)
%e A333175        (211)  (1132)
%e A333175               (2113)
%e A333175               (2311)
%e A333175               (3112)
%e A333175               (3211)
%e A333175 (End)
%p A333175 f:= n -> nops(numtheory:-factorset(n))!:
%p A333175 map(f, [$1..100]); # _Robert Israel_, Mar 12 2020
%t A333175 a[1] = 1; a[n_] := a[n] = Plus @@ (a[n/#[[1]]^#[[2]]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 100}]
%t A333175 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}]
%t A333175 Table[PrimeNu[n]!, {n, 1, 100}]
%Y A333175 Cf. A000142, A000961 (positions of 1's), A001221, A050363, A066504, A069513, A064372, A093320, A292586.
%Y A333175 Dominates A335451.
%Y A333175 Permutations of prime indices are A008480.
%Y A333175 Unsorted prime signature is A124010. Sorted prime signature is A118914.
%Y A333175 (1,2,1)-avoiding permutations of prime indices are A335449.
%Y A333175 (2,1,2)-avoiding permutations of prime indices are A335450.
%Y A333175 (1,2,1) or (2,1,2)-matching permutations of prime indices are A335460.
%Y A333175 (1,2,1) and (2,1,2)-matching permutations of prime indices are A335462.
%Y A333175 Cf. A056239, A112798, A181796, A333221, A335452, A335463, A335521.
%K A333175 nonn
%O A333175 1,6
%A A333175 _Ilya Gutkovskiy_, Mar 11 2020