I have been fascinated by irrational numbers ever since I first saw Euclid's proof that the square root of two cannot be expressed as a rational fraction. Among all the irrational numbers, the square roots of the integers occupy a special place, because they can be expressed as continued fractions with a repeating period of finite length. In fact, the square root of any rational number can be expressed as a regular periodic continued fraction. These expansions are not only periodic; they are, except for the value marking the end of each period, always palindromic. I discovered these facts for myself about 15 years ago, while playing around with an algorithm that allows one to construct the regular continued fraction expansion of √n, where n is any natural number. Later I read up on the subject, and learned that Joseph Louis Lagrange had plowed this ground pretty thoroughly about 250 years ago, ca.1766. I have worked through the proofs of the assertions made above (about periods, and palindromes), but I don't have the time or the inclination to put that stuff on my web site right now. Maybe some day. Today, I'm just aiming to present some concrete data: the periodic expansions of the square roots of the first 400 positive integers, listed in tabular form below. (Click here for a printable table in portable document format. If your monitor is in the old 4:3 aspect ratio -- generally 1024 x 768 -- or even some weird contraption like 5:4 & 1280 x 960 -- and the table doesn't look good in your browser, you may want to visit this alternative page.) In the following table "n" is an integer, "Continued Fraction" is the integer part of √n (set off by a semicolon) followed by one or two repetitions of the periodic part of the fraction, and "Period" is the length of the periodic part of the canonical continued fraction for √n. So, for example, the entry for n = 19 reads "4; 2, 1, 3, 1, 2, 8, ..." with a period of 6. Those entries represent the continued fraction where 4 is the integer part of √19, and the sequence of partial denominators {2, 1, 3, 1, 2, 8} repeats ad infinitum. |
n | Continued Fraction | Period | n | Continued Fraction | Period | n | Continued Fraction | Period | n | Continued Fraction | Period | n | Continued Fraction | Period |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | Perfect Square | 0 | 2 | 1; 2, 2, ... | 1 | 3 | 1; 1, 2, 1, 2, ... | 2 | 4 | Perfect Square | 0 | 5 | 2; 4, 4, ... | 1 |
6 | 2; 2, 4, 2, 4, ... | 2 | 7 | 2; 1, 1, 1, 4, ... | 4 | 8 | 2; 1, 4, 1, 4, ... | 2 | 9 | Perfect Square | 0 | 10 | 3; 6, 6, ... | 1 |
11 | 3; 3, 6, 3, 6, ... | 2 | 12 | 3; 2, 6, 2, 6, ... | 2 | 13 | 3; 1, 1, 1, 1, 6, ... | 5 | 14 | 3; 1, 2, 1, 6, ... | 4 | 15 | 3; 1, 6, 1, 6, ... | 2 |
16 | Perfect Square | 0 | 17 | 4; 8, 8, ... | 1 | 18 | 4; 4, 8, 4, 8, ... | 2 | 19 | 4; 2, 1, 3, 1, 2, 8, ... | 6 | 20 | 4; 2, 8, 2, 8, ... | 2 |
21 | 4; 1, 1, 2, 1, 1, 8, ... | 6 | 22 | 4; 1, 2, 4, 2, 1, 8, ... | 6 | 23 | 4; 1, 3, 1, 8, ... | 4 | 24 | 4; 1, 8, 1, 8, ... | 2 | 25 | Perfect Square | 0 |
26 | 5; 10, 10, ... | 1 | 27 | 5; 5, 10, 5, 10, ... | 2 | 28 | 5; 3, 2, 3, 10, ... | 4 | 29 | 5; 2, 1, 1, 2, 10, ... | 5 | 30 | 5; 2, 10, 2, 10, ... | 2 |
31 | 5; 1, 1, 3, 5, 3, 1, 1, 10, ... | 8 | 32 | 5; 1, 1, 1, 10, ... | 4 | 33 | 5; 1, 2, 1, 10, ... | 4 | 34 | 5; 1, 4, 1, 10, ... | 4 | 35 | 5; 1, 10, 1, 10, ... | 2 |
36 | Perfect Square | 0 | 37 | 6; 12, 12, ... | 1 | 38 | 6; 6, 12, 6, 12, ... | 2 | 39 | 6; 4, 12, 4, 12, ... | 2 | 40 | 6; 3, 12, 3, 12, ... | 2 |
41 | 6; 2, 2, 12, 2, 2, 12, ... | 3 | 42 | 6; 2, 12, 2, 12, ... | 2 | 43 | 6; 1, 1, 3, 1, 5, 1, 3, 1, 1, 12, ... | 10 | 44 | 6; 1, 1, 1, 2, 1, 1, 1, 12, ... | 8 | 45 | 6; 1, 2, 2, 2, 1, 12, ... | 6 |
46 | 6; 1,3,1,1,2,6,2,1,1,3,1,12,... | 12 | 47 | 6; 1, 5, 1, 12, ... | 4 | 48 | 6; 1, 12, 1, 12, ... | 2 | 49 | Perfect Square | 0 | 50 | 7; 14, 14, ... | 1 |
51 | 7; 7, 14, 7, 14, ... | 2 | 52 | 7; 4, 1, 2, 1, 4, 14, ... | 6 | 53 | 7; 3, 1, 1, 3, 14, ... | 5 | 54 | 7; 2, 1, 6, 1, 2, 14, ... | 6 | 55 | 7; 2, 2, 2, 14, ... | 4 |
56 | 7; 2, 14, 2, 14, ... | 2 | 57 | 7; 1, 1, 4, 1, 1, 14, ... | 6 | 58 | 7; 1, 1, 1, 1, 1, 1, 14, ... | 7 | 59 | 7; 1, 2, 7, 2, 1, 14, ... | 6 | 60 | 7; 1, 2, 1, 14, ... | 4 |
61 | 7; 1,4,3,1,2,2,1,3,4,1,14,... | 11 | 62 | 7; 1, 6, 1, 14, ... | 4 | 63 | 7; 1, 14, 1, 14, ... | 2 | 64 | Perfect Square | 0 | 65 | 8; 16, 16, ... | 1 |
66 | 8; 8, 16, 8, 16, ... | 2 | 67 | 8; 5, 2, 1, 1, 7, 1, 1, 2, 5, 16, ... | 10 | 68 | 8; 4, 16, 4, 16, ... | 2 | 69 | 8; 3, 3, 1, 4, 1, 3, 3, 16, ... | 8 | 70 | 8; 2, 1, 2, 1, 2, 16, ... | 6 |
71 | 8; 2, 2, 1, 7, 1, 2, 2, 16, ... | 8 | 72 | 8; 2, 16, 2, 16, ... | 2 | 73 | 8; 1, 1, 5, 5, 1, 1, 16, ... | 7 | 74 | 8; 1, 1, 1, 1, 16, ... | 5 | 75 | 8; 1, 1, 1, 16, ... | 4 |
76 | 8; 1,2,1,1,5,4,5,1,1,2,1,16,... | 12 | 77 | 8; 1, 3, 2, 3, 1, 16, ... | 6 | 78 | 8; 1, 4, 1, 16, ... | 4 | 79 | 8; 1, 7, 1, 16, ... | 4 | 80 | 8; 1, 16, 1, 16, ... | 2 |
81 | Perfect Square | 0 | 82 | 9; 18, 18, ... | 1 | 83 | 9; 9, 18, 9, 18, ... | 2 | 84 | 9; 6, 18, 6, 18, ... | 2 | 85 | 9; 4, 1, 1, 4, 18, ... | 5 |
86 | 9; 3, 1, 1, 1, 8, 1, 1, 1, 3, 18, ... | 10 | 87 | 9; 3, 18, 3, 18, ... | 2 | 88 | 9; 2, 1, 1, 1, 2, 18, ... | 6 | 89 | 9; 2, 3, 3, 2, 18, ... | 5 | 90 | 9; 2, 18, 2, 18, ... | 2 |
91 | 9; 1, 1, 5, 1, 5, 1, 1, 18, ... | 8 | 92 | 9; 1, 1, 2, 4, 2, 1, 1, 18, ... | 8 | 93 | 9; 1, 1, 1, 4, 6, 4, 1, 1, 1, 18, ... | 10 | 94 | 9; 1,2,3,1,1,5,1,8,1,5,1,1,3,2,1,18,... | 16 | 95 | 9; 1, 2, 1, 18, ... | 4 |
96 | 9; 1, 3, 1, 18, ... | 4 | 97 | 9; 1,5,1,1,1,1,1,1,5,1,18,... | 11 | 98 | 9; 1, 8, 1, 18, ... | 4 | 99 | 9; 1, 18, 1, 18, ... | 2 | 100 | Perfect Square | 0 |
101 | 10; 20, 20, ... | 1 | 102 | 10; 10, 20, 10, 20, ... | 2 | 103 | 10; 6,1,2,1,1,9,1,1,2,1,6,20,... | 12 | 104 | 10; 5, 20, 5, 20, ... | 2 | 105 | 10; 4, 20, 4, 20, ... | 2 |
106 | 10; 3, 2, 1, 1, 1, 1, 2, 3, 20, ... | 9 | 107 | 10; 2, 1, 9, 1, 2, 20, ... | 6 | 108 | 10; 2, 1, 1, 4, 1, 1, 2, 20, ... | 8 | 109 | 10; 2,3,1,2,4,1,6,6,1,4,2,1,3,2,20,... | 15 | 110 | 10; 2, 20, 2, 20, ... | 2 |
111 | 10; 1, 1, 6, 1, 1, 20, ... | 6 | 112 | 10; 1, 1, 2, 1, 1, 20, ... | 6 | 113 | 10; 1, 1, 1, 2, 2, 1, 1, 1, 20, ... | 9 | 114 | 10; 1, 2, 10, 2, 1, 20, ... | 6 | 115 | 10; 1, 2, 1, 1, 1, 1, 1, 2, 1, 20, ... | 10 |
116 | 10; 1, 3, 2, 1, 4, 1, 2, 3, 1, 20, ... | 10 | 117 | 10; 1, 4, 2, 4, 1, 20, ... | 6 | 118 | 10; 1, 6, 3, 2, 10, 2, 3, 6, 1, 20, ... | 10 | 119 | 10; 1, 9, 1, 20, ... | 4 | 120 | 10; 1, 20, 1, 20, ... | 2 |
121 | Perfect Square | 0 | 122 | 11; 22, 22, ... | 1 | 123 | 11; 11, 22, 11, 22, ... | 2 | 124 | 11; 7,2,1,1,1,3,1,4,1,3,1,1,1,2,7,22,... | 16 | 125 | 11; 5, 1, 1, 5, 22, ... | 5 |
126 | 11; 4, 2, 4, 22, ... | 4 | 127 | 11; 3,1,2,2,7,11,7,2,2,1,3,22,... | 12 | 128 | 11; 3, 5, 3, 22, ... | 4 | 129 | 11; 2, 1, 3, 1, 6, 1, 3, 1, 2, 22, ... | 10 | 130 | 11; 2, 2, 22, ... | 3 |
131 | 11; 2, 4, 11, 4, 2, 22, ... | 6 | 132 | 11; 2, 22, 2, 22, ... | 2 | 133 | 11; 1,1,7,5,1,1,1,2,1,1,1,5,7,1,1,22,... | 16 | 134 | 11; 1,1,2,1,3,1,10,1,3,1,2,1,1,22,... | 14 | 135 | 11; 1, 1, 1, 1, 1, 1, 1, 22, ... | 8 |
136 | 11; 1, 1, 1, 22, ... | 4 | 137 | 11; 1, 2, 2, 1, 1, 2, 2, 1, 22, ... | 9 | 138 | 11; 1, 2, 1, 22, ... | 4 | 139 | 11; 1,3,1,3,7,1,1,2,11,2,1,1,7,3,1,3,1,22,... | 18 | 140 | 11; 1, 4, 1, 22, ... | 4 |
141 | 11; 1, 6, 1, 22, ... | 4 | 142 | 11; 1, 10, 1, 22, ... | 4 | 143 | 11; 1, 22, 1, 22, ... | 2 | 144 | Perfect Square | 0 | 145 | 12; 24, 24, ... | 1 |
146 | 12; 12, 24, 12, 24, ... | 2 | 147 | 12; 8, 24, 8, 24, ... | 2 | 148 | 12; 6, 24, 6, 24, ... | 2 | 149 | 12; 4, 1, 5, 3, 3, 5, 1, 4, 24, ... | 9 | 150 | 12; 4, 24, 4, 24, ... | 2 |
151 | 12; 3,2,7,1,3,4,1,1,1,11,1,1,1,4,3,1,7,2,3,24,... | 20 | 152 | 12; 3, 24, 3, 24, ... | 2 | 153 | 12; 2, 1, 2, 2, 2, 1, 2, 24, ... | 8 | 154 | 12; 2, 2, 3, 1, 2, 1, 3, 2, 2, 24, ... | 10 | 155 | 12; 2, 4, 2, 24, ... | 4 |
156 | 12; 2, 24, 2, 24, ... | 2 | 157 | 12; 1,1,7,1,5,2,1,1,1,1,2,5,1,7,1,1,24,... | 17 | 158 | 12; 1, 1, 3, 12, 3, 1, 1, 24, ... | 8 | 159 | 12; 1, 1, 1, 1, 3, 1, 1, 1, 1, 24, ... | 10 | 160 | 12; 1, 1, 1, 5, 1, 1, 1, 24, ... | 8 |
161 | 12; 1, 2, 4, 1, 2, 1, 4, 2, 1, 24, ... | 10 | 162 | 12; 1, 2, 1, 2, 12, 2, 1, 2, 1, 24, ... | 10 | 163 | 12; 1,3,3,2,1,1,7,1,11,1,7,1,1,2,3,3,1,24,... | 18 | 164 | 12; 1, 4, 6, 4, 1, 24, ... | 6 | 165 | 12; 1, 5, 2, 5, 1, 24, ... | 6 |
166 | 12; 1,7,1,1,1,2,4,1,3,2,12,2,3,1,4,2,1,1,1,7,1,24,... | 22 | 167 | 12; 1, 11, 1, 24, ... | 4 | 168 | 12; 1, 24, 1, 24, ... | 2 | 169 | Perfect Square | 0 | 170 | 13; 26, 26, ... | 1 |
171 | 13; 13, 26, 13, 26, ... | 2 | 172 | 13; 8,1,2,2,1,1,3,6,3,1,1,2,2,1,8,26,... | 16 | 173 | 13; 6, 1, 1, 6, 26, ... | 5 | 174 | 13; 5, 4, 5, 26, ... | 4 | 175 | 13; 4, 2, 1, 2, 4, 26, ... | 6 |
176 | 13; 3, 1, 3, 26, ... | 4 | 177 | 13; 3, 3, 2, 8, 2, 3, 3, 26, ... | 8 | 178 | 13; 2, 1, 12, 1, 2, 26, ... | 6 | 179 | 13; 2,1,1,1,3,5,13,5,3,1,1,1,2,26,... | 14 | 180 | 13; 2, 2, 2, 26, ... | 4 |
181 | 13; 2,4,1,8,6,1,1,1,1,2,2,1,1,1,1,6,8,1,4,2,26,... | 21 | 182 | 13; 2, 26, 2, 26, ... | 2 | 183 | 13; 1, 1, 8, 1, 1, 26, ... | 6 | 184 | 13; 1,1,3,2,1,2,1,2,3,1,1,26,... | 12 | 185 | 13; 1, 1, 1, 1, 26, ... | 5 |
186 | 13; 1, 1, 1, 3, 4, 3, 1, 1, 1, 26, ... | 10 | 187 | 13; 1, 2, 13, 2, 1, 26, ... | 6 | 188 | 13; 1, 2, 2, 6, 2, 2, 1, 26, ... | 8 | 189 | 13; 1, 2, 1, 26, ... | 4 | 190 | 13; 1,3,1,1,1,2,2,2,1,1,1,3,1,26,... | 14 |
191 | 13; 1,4,1,1,3,2,2,13,2,2,3,1,1,4,1,26,... | 16 | 192 | 13; 1, 5, 1, 26, ... | 4 | 193 | 13; 1,8,3,2,1,3,3,1,2,3,8,1,26,... | 13 | 194 | 13; 1, 12, 1, 26, ... | 4 | 195 | 13; 1, 26, 1, 26, ... | 2 |
196 | Perfect Square | 0 | 197 | 14; 28, 28, ... | 1 | 198 | 14; 14, 28, 14, 28, ... | 2 | 199 | 14; 9,2,1,2,2,5,4,1,1,13,1,1,4,5,2,2,1,2,9,28,... | 20 | 200 | 14; 7, 28, 7, 28, ... | 2 |
201 | 14; 5,1,1,1,2,1,8,1,2,1,1,1,5,28,... | 14 | 202 | 14; 4, 1, 2, 2, 1, 4, 28, ... | 7 | 203 | 14; 4, 28, 4, 28, ... | 2 | 204 | 14; 3, 1, 1, 6, 1, 1, 3, 28, ... | 8 | 205 | 14; 3, 6, 2, 4, 1, 6, 3, 28, ... | 8 |
206 | 14; 2, 1, 5, 14, 5, 1, 2, 28, ... | 8 | 207 | 14; 2, 1, 1, 2, 1, 1, 2, 28, ... | 8 | 208 | 14; 2, 2, 1, 2, 2, 28, ... | 6 | 209 | 14; 2, 5, 3, 2, 3, 5, 2, 28, ... | 8 | 210 | 14; 2, 28, 2, 28, ... | 2 |
211 | 14; 1,1,9,5,1,2,2,1,1,4,3,1,13,1,3,4,1,1,2,2,1,5,9,1,1,28,... | 26 | 212 | 14; 1,1,3,1,1,1,6,1,1,1,3,1,1,28,... | 14 | 213 | 14; 1,1,2,6,1,8,1,6,2,1,1,28,... | 12 | 214 | 14; 1,1,1,2,3,1,4,9,1,1,5,3,14,3,5,1,1,9,4,1,3,2,1,1,1,28,... | 26 | 215 | 14; 1, 1, 1, 28, ... | 4 |
216 | 14; 1, 2, 3, 2, 1, 28, ... | 6 | 217 | 14; 1,2,1,2,1,1,9,4,9,1,1,2,1,2,1,28,... | 16 | 218 | 14; 1, 3, 3, 1, 28, ... | 5 | 219 | 14; 1, 3, 1, 28, ... | 4 | 220 | 14; 1, 4, 1, 28, ... | 4 |
221 | 14; 1, 6, 2, 6, 1, 28, ... | 6 | 222 | 14; 1, 8, 1, 28, ... | 4 | 223 | 14; 1, 13, 1, 28, ... | 4 | 224 | 14; 1, 28, 1, 28, ... | 2 | 225 | Perfect Square | 0 |
226 | 15; 30, 30, ... | 1 | 227 | 15; 15, 30, 15, 30, ... | 2 | 228 | 15; 10, 30, 10, 30, ... | 2 | 229 | 15; 7, 1, 1, 7, 30, ... | 5 | 230 | 15; 6, 30, 6, 30, ... | 4 |
231 | 15; 5, 30, 5, 30, ... | 2 | 232 | 15; 4, 3, 7, 3, 4, 30, ... | 6 | 233 | 15; 3, 1, 3, 1, 1, 1, 1, 3, 1, 3, 30,... | 11 | 234 | 15; 3, 2, 1, 2, 1, 2, 3, 30, ... | 8 | 235 | 15; 3, 30, 3, 30, ... | 2 |
236 | 15; 2,1,3,5,1,6,1,5,3,1,2,30,... | 12 | 237 | 15; 2, 1, 1, 7, 10, 7, 1, 1, 2, 30, ... | 10 | 238 | 15; 2, 2, 1, 14, 1, 2, 2, 30, ... | 8 | 239 | 15; 2,5,1,2,4,15,4,2,1,5,2,30,... | 12 | 240 | 15; 2, 30, 2, 30, ... | 2 |
241 | 15; 1,1,9,1,5,3,3,1,1,3,3,5,1,9,1,1,30,... | 17 | 242 | 15; 1, 1, 3, 1, 14, 1, 3, 1, 1, 30, ... | 10 | 243 | 15; 1, 1, 2, 3, 15, 3, 2, 1, 1, 30, ... | 10 | 244 | 15; 1,1,1,1,1,2,1,5,1,1,9,1,6,1,9,1,1,5,1,2,1,1,1,1,1,30,... | 26 | 245 | 15; 1, 1, 1, 7, 6, 7, 1, 1, 1, 30, ... | 10 |
246 | 15; 1, 1, 2, 3, 15, 3, 2, 1, 1, 30, ... | 10 | 247 | 15; 1,2,1,1,9,1,9,1,1,2,1,30,... | 12 | 248 | 15; 1, 2, 1, 30, ... | 4 | 249 | 15; 1,3,1,1,5,1,3,10,3,1,5,1,1,3,1,30,... | 16 | 250 | 15; 1, 4, 3, 3, 4, 1, 30, ... | 7 |
251 | 15; 1,5,2,1,2,2,15,2,2,1,2,5,1,30,... | 14 | 252 | 15; 1, 6, 1, 30, ... | 4 | 253 | See Note 1a, infra | 22 | 254 | 15; 1, 14, 1, 30, ... | 4 | 255 | 15; 1, 30, 1, 30, ... | 2 |
256 | Perfect Square | 0 | 257 | 16; 32, 32, ... | 1 | 258 | 16; 16, 32, 16, 32, ... | 2 | 259 | 16; 10, 1, 2, 3, 4, 3, 2, 1, 10, 32, ... | 10 | 260 | 16; 8, 32, 8, 32, ... | 2 |
261 | 16; 6,2,3,7,1,3,1,2,1,3,1,7,3,2,6,32,... | 16 | 262 | 16; 5,2,1,2,1,10,16,101,2,1,2,5,32,... | 14 | 263 | 16; 4,1,1,1,1,15,1,1,1,1,4,32,... | 12 | 264 | 16; 4, 32, 4, 32, ... | 2 | 265 | 16; 3, 1, 1, 2, 2, 1, 1, 3, 32, ... | 9 |
266 | 16; 3, 4, 3, 32, ... | 4 | 267 | 16; 2, 1, 15, 1, 2, 32, ... | 6 | 268 | See Note 2a, infra | 20 | 269 | 16; 2, 2, 32, ... | 3 | 270 | 16; 2, 3, 6, 3, 2, 32, ... | 6 |
271 | 16; 2,6,10,1,4,1,1,2,1,2,1,15,1,2,1,2,1,1,4,1,10,6,2,32,... | 24 | 272 | 16; 2, 32, 2, 32, ... | 2 | 273 | 16; 1, 1, 10, 1, 1, 32, ... | 6 | 274 | 16; 1, 1, 4, 4, 1, 1, 32, ... | 7 | 275 | 16; 1, 1, 2, 1, 1, 32, ... | 6 |
276 | 16; 1,1,1,1,2,2,2,1,1,1,1,32, ... | 12 | 277 | See Note 2b, infra | 21 | 278 | 16; 1, 2, 16, 2, 1, 32, ... | 6 | 279 | 16; 1, 2, 2, 1, 2, 2, 1, 32, ... | 8 | 280 | 16; 1, 2, 1, 2, 1, 32, ... | 6 |
281 | 16; 1,3,4,1,1,6,6,1,1,4,3,1,32,... | 13 | 282 | 16; 1, 3, 1, 4, 1, 3, 1, 32, ... | 8 | 283 | 16; 1, 2, 1, 2, 1, 32, ... | 6 | 284 | 16; 1,5,1,3,2,1,4,8,4,1,2,3,1,5,1,32,... | 16 | 285 | 16; 1, 7, 2, 7, 1, 32, ... | 6 |
286 | 16; 1, 10, 3, 3, 2, 3, 3, 10, 1, 32, ... | 10 | 287 | 16; 1, 15, 1, 32, ... | 4 | 288 | 16; 1, 32, 1, 32, ... | 2 | 289 | Perfect Square | 0 | 290 | 17; 34, 34, ... | 1 |
291 | 17; 17, 34, 17, 34, ... | 2 | 292 | 17; 11, 2, 1, 3, 8, 3, 1, 2, 11, 34, ... | 10 | 293 | 17; 8, 1, 1, 8, 34, ... | 5 | 294 | 17; 6, 1, 4, 1, 6, 34, ... | 6 | 295 | 17; 5,1,2,3,2,6,2,3,2,1,5,34,... | 12 |
296 | 17; 4, 1, 7, 1, 34, ... | 6 | 297 | 17; 4, 3, 1, 1, 2, 1, 1, 3, 4, 34, ... | 10 | 298 | 17; 3,1,4,5,1,1,5,4,1,3,34,... | 11 | 299 | 17; 3, 2, 3, 34, ... | 4 | 300 | 17; 3, 8, 3, 34, ... | 4 |
301 | 17; 2,1,6,3,1,2,2,1,1,8,11,2,4,2,11,8,1,1,2,2,1,3,6,1,2,34,... | 26 | 302 | 17; 2,1,1,1,4,2,1,16,1,2,4,1,1,1,2,34,... | 16 | 303 | 17; 2, 2, 5, 2, 2, 34, ... | 6 | 304 | 17; 2,3,2,1,1,1,1,1,2,3,2,34,... | 12 | 305 | 17; 2, 6, 2, 34, ... | 4 |
306 | 17; 2, 34, 2, 34, ... | 2 | 307 | 17; 1,1,11,5,1,3,17,3,1,5,11,1,1,34,... | 14 | 308 | 17; 1, 1, 4, 1, 1, 34, ... | 6 | 309 | 17; 1,1,2,1,2,4,1,1,1,8,6,1,10,1,6,8,1,1,1,4,2,1,2,1,1,34,... | 26 | 310 | See note 3a, infra | 16 |
311 | 17; 1,1,1,2,1,6,3,17,3,6,1,2,1,1,1,34,... | 16 | 312 | 17; 1, 1, 1, 34, ... | 4 | 313 | 17; 1,2,4,11,1,1,3,2,2,3,1,1,11,4,2,1,34,... | 17 | 314 | 17; 1, 2, 1, 1, 2, 1, 34, ... | 7 | 315 | 17; 1, 2, 1, 34, ... | 4 |
316 | 17; 1, 3, 2, 8, 2, 3, 1, 34, ... | 8 | 317 | 17; 1,4,8,1,2,2,1,8,4,1,34,... | 11 | 318 | 17; 1, 4, 1, 34, ... | 4 | 319 | 17; 1,6,5,1,4,3,1,3,4,1,5,6,1,34,... | 14 | 320 | 17; 1, 7, 1, 34, ... | 4 |
321 | 17; 1, 10, 1, 34, ... | 4 | 322 | 17; 1, 16, 1, 34, ... | 4 | 323 | 17; 1, 34, 1, 34, ... | 2 | 324 | Perfect Square | 0 | 325 | 18; 36, 36, ... | 1 |
326 | 18; 18, 36, 18, 16, ... | 2 | 327 | 18; 12, 36, 12, 36, ... | 2 | 328 | 18; 9, 36, 9, 36, ... | 2 | 329 | 18; 7, 1, 1, 7, 30, ... | 5 | 330 | 18; 6, 30, 6, 30, ... | 4 |
331 | See Note 4a, infra | 34 | 332 | 18; 4, 1, 1, 8, 1, 1, 4, 36, ... | 8 | 333 | 18; 4, 36, 4, 36,... | 2 | 334 | See Note 4b, infra | 28 | 335 | 18; 3, 3, 3, 36, ... | 4 |
336 | 18; 3, 36, 3, 36,... | 2 | 337 | See Note 4c, infra | 19 | 338 | 18; 2, 1, 1, 2, 36, ... | 5 | 339 | 18; 2, 2, 2, 1, 17, 1, 2, 2, 2, 36, ... | 10 | 340 | 18; 2,3,1,1,1,1,8,1,1,1,1,3,2,36,... | 14 |
341 | 18; 2,6,1,8,2,1,2,1,2,8,1,6,2,36,... | 14 | 342 | 18; 2, 36, 2, 36, ... | 2 | 343 | 18; 1,1,11,1,5,3,1,17,1,3,5,1,11,1,1,36, ... | 16 | 344 | 18; 1, 1, 4, 1, 3, 1, 4, 1, 1, 36, ... | 10 | 345 | 18; 1, 1, 2, 1, 6, 1, 2, 1, 1, 36, ... | 10 |
346 | 18; 1, 1, 1, 1, 36, ... | 5 | 347 | 18; 1,1,1,2,4,1,17,1,4,2,1,1,1,36,... | 14 | 348 | 18; 1, 1, 1, 8, 1, 1, 1, 36, ... | 8 | 349 | 18; 1, 2, 7, 7, 2, 1, 36, ... | 7 | 350 | 18; 1,2,2,2,1,36, ... | 6 |
351 | 18; 1,2,1,3,2,2,2,3,1,2,1,36,... | 12 | 352 | 18; 1, 3, 5, 9, 5, 3, 1, 36, ... | 8 | 353 | 18; 1,3,1,2,1,1,1,1,1,1,2,1,3,1,36,... | 15 | 354 | 18; 1, 4, 2, 2, 18, 2, 2, 4, 1, 36, ... | 10 | 355 | 18; 1,5,3,3,1,6,2,3,3,5,1,36, ... | 12 |
356 | 18; 1,6,1,1,2,1,8,1,2,1,1,6,1,36,... | 14 | 357 | 18; 1, 8, 2, 8, 1, 36, ... | 6 | 358 | See Note 4d, infra | 26 | 359 | 18; 1, 17, 1, 36, ... | 4 | 360 | 18; 1, 36, 1, 36, ... | 2 |
361 | Perfect Square | 0 | 362 | 19; 38, 38, ... | 14 | 363 | 19; 19, 38, 19, 38, ... | 2 | 364 | 19; 12,1,2,3,1,8,1,3,2,1,12,38,... | 12 | 365 | 19; 9, 1, 1, 9, 38, ... | 5 |
366 | 19; 7,1,1,1,2,12,2,1,1,1,7,38, ... | 12 | 367 | See Note 5a, infra | 20 | 368 | 19; 5, 2, 5, 38, ... | 5 | 369 | 19; 4,1,3,2,7,4,7,2,3,1,4,38, ... | 12 | 370 | 19; 4, 4, 38, ... | 3 |
371 | 19; 3, 1, 4, 1, 3, 38, ... | 6 | 372 | 19; 3, 2, 12, 2, 3, 38, ... | 6 | 373 | 19; 3, 5, 5, 3, 38, ... | 5 | 374 | 19; 2, 1, 18, 1, 2, 38, ... | 6 | 375 | 19; 2, 1, 2, 1, 5, 1, 2, 1, 2, 38, ... | 10 |
376 | 19; 2,1,1,3,1,2,2,4,2,2,1,3,1,1,2,38, ... | 16 | 377 | 19; 2, 2, 2, 2, 38, ... | 5 | 378 | 19; 2, 3, 1, 4, 1, 3, 2, 38, ... | 8 | 379 | See Note 5b, infra | 30 | 380 | 19; 2, 38, ... | 2 |
381 | 19; 1, 1, 12, 1, 1, 38, ... | 8 | 382 | See Note 5c, infra | 24 | 383 | 19; 1, 1, 3, 19, 3, 1, 1, 38, ... | 8 | 384 | 19; 1, 1, 2, 9, 2, 1, 1, 38, ... | 8 | 385 | See Note 5d, infra | 16 |
386 | 19; 1,1,1,4,1,18,1,4,1,1,38,... | 12 | 387 | 19; 1, 2, 19, 2, 1, 38, ... | 6 | 388 | 19; 1,2,3,4,12,1,8,1,12,4,3,2,1,38,... | 14 | 389 | 19; 1, 2, 1, 1, 1, 1, 2, 1, 38, ... | 9 | 390 | 19; 1, 2, 1, 38, ... | 4 |
391 | 19; 1,3,2,2,1,1,2,19,2,1,1,2,2,3,1,38,... | 16 | 392 | 19; 1, 3, 1, 38, ... | 4 | 393 | See Note 5e, infra | 20 | 394 | 19; 1,5,1,1,1,3,1,3,5,2,2,5,3,1,3,1,1,1,5,1,38,... | 21 | 395 | 19; 1, 6, 1, 38, ... | 4 |
396 | 19; 1, 8, 1, 38, ... | 4 | 397 | See Note 5f, infra | 21 | 398 | 19; 1, 18, 1, 38,... | 4 | 399 | 19; 1, 38, 1, 38, ... | 2 | 400 | Perfect Square | 0 |
Notes ... 1a: 253 15; 1,9,1,1,1,2,1,7,4,2,2,2,4,7,1,2,1,1,1,9,1,30,... 5a: 367 19; 6,2,1,3,1,1,2,1,12,19,12,1,2,1,1,3,1,2,6,38,... 2a: 268 16; 2,1,2,3,3,1,3,1,10,8,10,1,3,1,3,3,2,1,2,32,... 5b: 379 19; 2,7,3,2,2,6,12,1,4,1,1,1,3,4,19,4,3,1,1,1,4,1,12,6,2,2,3,7,2,38,... 2b: 277 16; 1,1,1,4,10,1,7,2,2,3,3,2,2,7,1,10,4,1,1,1,32,... 5c: 382 19; 1,1,5,12,1,5,1,1,2,3,1,18,1,3,2,1,1,5,1,12,5,1,1,38,... 3a: 310 17; 1,1,1,1,5,3,1,2,1,3,5,1,1,1,1,34,... 5d: 385 19; 1,1,1,1,1,3,1,2,1,3,1,1,1,1,1,38,... 4a: 331 18; 5,5,1,6,2,3,1,1,2,1,2,1,11,2,1,1,17,1,1,2,11,1,2,1,2,1,1,3,2,6,1,5,5,36,... 5e: 393 19; 1,4,1,2,4,1,1,1,1,12,1,1,1,1,4,2,1,4,1,38,... 4b: 334 18; 3,1,1,1,2,5,1,2,2,11,1,3,7,18,7,3,1,11,2,2,1,5,2,1,1,1,3,36,... 5f: 397 19; 1,12,3,4,9,1,2,1,2,1,1,2,1,2,1,9,4,3,12,1,38,... 4c: 337 18; 2,1,3,1,11,2,4,1,3,3,1,4,2,11,1,3,1,2,36,... 4d: 358 18; 1,11,1,1,1,3,1,1,4,1,5,2,18,2,5,1,4,1,1,3,1,1,1,11,1,36,...