Buggy python submission (verdict: Time Limit Exceeded):
import math
import collections

k = int(input())
gcd_table = [[0] * k for i in range(k)]
for i in range(1, k+1):
    for j in range(i, k+1):
        gcd_table[i-1][j-1] = math.gcd(i, j)

ans = 0
combination_list = [0, 1, 3, 6]
for p in range(1, k+1):
    for q in range(p, k+1):
        if gcd_table[p-1][q-1] == 0:
            tmp_gcd = math.gcd(p, q)
            gcd_table[p-1][q-1] = tmp_gcd
        else:
            tmp_gcd = gcd_table[p-1][q-1]
        for r in range(q, k+1):
            if gcd_table[q-1][r-1] == 0:
                tmp_gcd_2 = math.gcd(q, r)
                gcd_table[q-1][r-1] = tmp_gcd_2
            else:
                tmp_gcd_2 = gcd_table[q-1][r-1]
            unique_num = len(collections.Counter([p, q, r]))
            ans += math.gcd(tmp_gcd, tmp_gcd_2) * combination_list[unique_num]
print(ans)

Fixed submission:
import math
import collections

k = int(input())
gcd_table = [[0] * k for i in range(k)]
for i in range(1, k+1):
    for j in range(i, k+1):
        gcd_table[i-1][j-1] = math.gcd(i, j)

ans = 0
combination_list = [0, 1, 3, 6]
for p in range(1, k+1):
    for q in range(p, k+1):
        if gcd_table[p-1][q-1] == 0:
            tmp_gcd = math.gcd(p, q)
            gcd_table[p-1][q-1] = tmp_gcd
        else:
            tmp_gcd = gcd_table[p-1][q-1]
        for r in range(q, k+1):
            if gcd_table[q-1][r-1] == 0:
                tmp_gcd_2 = math.gcd(q, r)
                gcd_table[q-1][r-1] = tmp_gcd_2
            else:
                tmp_gcd_2 = gcd_table[q-1][r-1]
            unique_num = len(collections.Counter([p, q, r]))
            ans += math.gcd(tmp_gcd, tmp_gcd_2) * combination_list[unique_num]
print(ans)

Buggy python submission (verdict: Time Limit Exceeded):
import math
import collections

k = int(input())
gcd_table = [[0] * k for i in range(k)]
ans = 0
combination_list = [0, 1, 3, 6]
for p in range(1, k+1):
    for q in range(p, k+1):
        if gcd_table[p-1][q-1] == 0:
            tmp_gcd = math.gcd(p, q)
            gcd_table[p-1][q-1] = tmp_gcd
        else:
            tmp_gcd = gcd_table[p-1][q-1]
        for r in range(q, k+1):
            if gcd_table[q-1][r-1] == 0:
                tmp_gcd_2 = math.gcd(q, r)
                gcd_table[q-1][r-1] = tmp_gcd_2
            else:
                tmp_gcd_2 = gcd_table[q-1][r-1]
            unique_num = len(collections.Counter([p, q, r]))
            ans += math.gcd(tmp_gcd, tmp_gcd_2) * combination_list[unique_num]
print(ans)

Fixed submission:
import math
import collections

k = int(input())
gcd_table = [[0] * k for i in range(k)]
for i in range(1, k+1):
    for j in range(i, k+1):
        gcd_table[i-1][j-1] = math.gcd(i, j)

ans = 0
combination_list = [0, 1, 3, 6]
for p in range(1, k+1):
    for q in range(p, k+1):
        if gcd_table[p-1][q-1] == 0:
            tmp_gcd = math.gcd(p, q)
            gcd_table[p-1][q-1] = tmp_gcd
        else:
            tmp_gcd = gcd_table[p-1][q-1]
        for r in range(q, k+1):
            if gcd_table[q-1][r-1] == 0:
                tmp_gcd_2 = math.gcd(q, r)
                gcd_table[q-1][r-1] = tmp_gcd_2
            else:
                tmp_gcd_2 = gcd_table[q-1][r-1]
            unique_num = len(collections.Counter([p, q, r]))
            ans += math.gcd(tmp_gcd, tmp_gcd_2) * combination_list[unique_num]
print(ans)