diff --git a/stdlib/Random/src/misc.jl b/stdlib/Random/src/misc.jl
index d24fe9c457ccf..19689613d387f 100644
--- a/stdlib/Random/src/misc.jl
+++ b/stdlib/Random/src/misc.jl
@@ -255,9 +255,12 @@ shuffle(a::AbstractArray) = shuffle(GLOBAL_RNG, a)
     randperm([rng=GLOBAL_RNG,] n::Integer)
 
 Construct a random permutation of length `n`. The optional `rng`
-argument specifies a random number generator (see [Random Numbers](@ref)).
-To randomly permute an arbitrary vector, see [`shuffle`](@ref)
-or [`shuffle!`](@ref).
+argument specifies a random number generator (see [Random
+Numbers](@ref)). The element type of the result is the same as the type
+of `n`.
+
+To randomly permute an arbitrary vector, see [`shuffle`](@ref) or
+[`shuffle!`](@ref).
 
 # Examples
 ```jldoctest
@@ -269,7 +272,7 @@ julia> randperm(MersenneTwister(1234), 4)
  3
 ```
 """
-randperm(r::AbstractRNG, n::Integer) = randperm!(r, Vector{Int}(undef, n))
+randperm(r::AbstractRNG, n::T) where {T <: Integer} = randperm!(r, Vector{T}(undef, n))
 randperm(n::Integer) = randperm(GLOBAL_RNG, n)
 
 """
@@ -317,6 +320,7 @@ randperm!(a::Array{<:Integer}) = randperm!(GLOBAL_RNG, a)
 
 Construct a random cyclic permutation of length `n`. The optional `rng`
 argument specifies a random number generator, see [Random Numbers](@ref).
+The element type of the result is the same as the type of `n`.
 
 # Examples
 ```jldoctest
@@ -330,7 +334,7 @@ julia> randcycle(MersenneTwister(1234), 6)
  2
 ```
 """
-randcycle(r::AbstractRNG, n::Integer) = randcycle!(r, Vector{Int}(undef, n))
+randcycle(r::AbstractRNG, n::T) where {T <: Integer} = randcycle!(r, Vector{T}(undef, n))
 randcycle(n::Integer) = randcycle(GLOBAL_RNG, n)
 
 """
diff --git a/stdlib/Random/test/runtests.jl b/stdlib/Random/test/runtests.jl
index 83e4b44e83697..56318d3e2c162 100644
--- a/stdlib/Random/test/runtests.jl
+++ b/stdlib/Random/test/runtests.jl
@@ -485,18 +485,26 @@ let mta = MersenneTwister(42), mtb = MersenneTwister(42)
     @test sort!(randperm(10)) == sort!(shuffle(1:10)) == 1:10
     @test randperm(mta,big(10)) == randperm(mtb,big(10)) # cf. #16376
     @test randperm(0) == []
-    @test eltype(randperm(UInt(1))) === Int
     @test_throws ErrorException randperm(-1)
 
+    let p = randperm(UInt16(12))
+        @test typeof(p) ≡ Vector{UInt16}
+        @test sort!(p) == 1:12
+    end
+
     A, B = Vector{Int}(undef, 10), Vector{Int}(undef, 10)
     @test randperm!(mta, A) == randperm!(mtb, B)
     @test randperm!(A) === A
 
     @test randcycle(mta,10) == randcycle(mtb,10)
-    @test eltype(randcycle(UInt(1))) === Int
     @test randcycle!(mta, A) == randcycle!(mtb, B)
     @test randcycle!(A) === A
 
+    let p = randcycle(UInt16(10))
+        @test typeof(p) ≡ Vector{UInt16}
+        @test sort!(p) == 1:10
+    end
+
     @test sprand(mta,1,1,0.9) == sprand(mtb,1,1,0.9)
     @test sprand(mta,10,10,0.3) == sprand(mtb,10,10,0.3)
 end