To improve stickiness and elusiveness to organic solvents by forming a 1-butene homopolymer or a copolymer with propylene or an alkenyl hydrocarbon of at least 5 carbon atoms which homopolymer or copolymer has a number average molecular weight of not less than a specific value and is amorphous.

This olefin polymer is a 1-butene homopolymer or a copolymer with propylene or an alkenyl hydrocarbon of at least 5 carbon atoms and is an olefin polymer comprising an amorphous polymer having a number average molecular weight, calculated in terms of polystyrene, of not less than 200,000 and a molecular weight distribution of not more than 3.0. The catalyst for polymerization is obtained by reacting an organic aluminum oxy compound of formula II or III and 0.1-3.0 mol, against 1 mol of Al atom of the compound, of water and then reacting it with 1 mol, against 10,000-20,000 mol of Al atom, of a transition metal atom of formula I. In the formulas, R is methyl or isobutyl; m is 1-50; M is a transition metal atom of the group IV in the periodic table; X and Y are each H, a halogen or the like; and T stands for a 1-20C bivalent covalent crosslinking group.