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
Keywords
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.
(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.
Programs
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Maple
f:= n -> nops(numtheory:-factorset(n))!: map(f, [$1..100]); # Robert Israel, Mar 12 2020
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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}]
Comments