Per the Social Security Administration website, SSA does not consider the suffix to be part of the legal name.
https://secure.ssa.gov/apps10/poms.nsf/lnx/0110212001
https://secure.ssa.gov/apps10/poms.nsf/lnx/0110205120
https://secure.ssa.gov/apps10/poms.nsf/lnx/0110212001
https://secure.ssa.gov/apps10/poms.nsf/lnx/0110205120