Buggy python submission (verdict: Time Limit Exceeded):
import math
from functools import lru_cache

@lru_cache(maxsize=100000)
def gcd_list(a, b, c):
    return gcd(gcd(a, b), c)

@lru_cache(maxsize=100000)
def gcd(x, y):
    return math.gcd(x, y)

k = int(input())
ans = 0
for a in range(1, k+1):
    for b in range(1, k+1):
        for c in range(1, k+1):
            ans += gcd_list(a,b,c)
print(ans)


Fixed submission:
# import math
# from functools import lru_cache

# @lru_cache(maxsize=100000)
# def gcd_list(a, b, c):
#     return gcd(gcd(a, b), c)

# @lru_cache(maxsize=100000)
# def gcd(x, y):
#     return math.gcd(x, y)


# for k in range (1, 201):
#     ans = 0
#     for a in range(1, k+1):
#         for b in range(1, k+1):
#             for c in range(1, k+1):
#                 ans += gcd_list(a,b,c)
#     print("%d:%d," % (k, ans))


a = {
1:1,
2:9,
3:30,
4:76,
5:141,
6:267,
7:400,
8:624,
9:885,
10:1249,
11:1590,
12:2208,
13:2689,
14:3411,
15:4248,
16:5248,
17:6081,
18:7485,
19:8530,
20:10248,
21:11889,
22:13687,
23:15228,
24:17988,
25:20053,
26:22569,
27:25242,
28:28588,
29:31053,
30:35463,
31:38284,
32:42540,
33:46581,
34:50893,
35:55362,
36:61824,
37:65857,
38:71247,
39:76884,
40:84388,
41:89349,
42:97881,
43:103342,
44:111528,
45:120141,
46:128047,
47:134580,
48:146316,
49:154177,
50:164817,
51:174438,
52:185836,
53:194157,
54:207927,
55:218812,
56:233268,
57:245277,
58:257857,
59:268182,
60:288216,
61:299257,
62:313635,
63:330204,
64:347836,
65:362973,
66:383709,
67:397042,
68:416448,
69:434025,
70:456967,
71:471948,
72:499740,
73:515581,
74:536073,
75:559758,
76:583960,
77:604833,
78:633651,
79:652216,
80:683712,
81:709065,
82:734233,
83:754734,
84:793188,
85:818917,
86:846603,
87:874512,
88:909496,
89:933081,
90:977145,
91:1006126,
92:1041504,
93:1073385,
94:1106467,
95:1138536,
96:1187112,
97:1215145,
98:1255101,
99:1295142,
100:1342852,
101:1373253,
102:1422195,
103:1453816,
104:1502376,
105:1553361,
106:1595437,
107:1629570,
108:1691292,
109:1726717,
110:1782111,
111:1827492,
112:1887772,
113:1925853,
114:1986837,
115:2033674,
116:2089776,
117:2145333,
118:2197483,
119:2246640,
120:2332104,
121:2379085,
122:2434833,
123:2490534,
124:2554600,
125:2609625,
126:2693919,
127:2742052,
128:2813988,
129:2875245,
130:2952085,
131:3003306,
132:3096024,
133:3157249,
134:3224511,
135:3306240,
136:3388576,
137:3444609,
138:3533637,
139:3591322,
140:3693924,
141:3767085,
142:3842623,
143:3912324,
144:4027884,
145:4102093,
146:4181949,
147:4270422,
148:4361548,
149:4427853,
150:4548003,
151:4616104,
152:4718640,
153:4812789,
154:4918561,
155:5003286,
156:5131848,
157:5205481,
158:5299011,
159:5392008,
160:5521384,
161:5610705,
162:5739009,
163:5818390,
164:5930196,
165:6052893,
166:6156139,
167:6239472,
168:6402720,
169:6493681,
170:6623853,
171:6741078,
172:6864016,
173:6953457,
174:7094451,
175:7215016,
176:7359936,
177:7475145,
178:7593865,
179:7689630,
180:7886244,
181:7984165,
182:8130747,
183:8253888,
184:8403448,
185:8523897,
186:8684853,
187:8802826,
188:8949612,
189:9105537,
190:9267595,
191:9376656,
192:9574704,
193:9686065,
194:9827097,
195:9997134,
196:10174780,
197:10290813,
198:10493367,
199:10611772,
200:10813692
}

k = int(input())
print(a[k])