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Any triangle-based pyramid will do. If the sides are all the same length it's then called a tetrahedron.
A pyramid is a generic term for a 3-D polyhedron which has a polygonal base and triangular lateral faces that meet at an apex above the base. If the polygonal base has n sides, the pyramid has n+1 faces, n+1 vertices and 2n edges. A pyramid can, therefore have 4 or more vertices. So, there is bound to be a pyramid that will have the same number of vertices as a triangular prism (6).
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A rectangular prism (cuboid) and a hexagon-based pyramid, for example, both have 12 edges. Of the five Platonic solids, an octahedron and a cube each have 12 edges.