Continued Fraction Expansions of Integral Square Roots

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 399 positive integers, listed in tabular form below. (Click here for a printable table in portable document format.)

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

Canonical Continued Fraction for Sqrt(19)

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
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

   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,...
   2a: 268 16; 2,1,2,3,3,1,3,1,10,8,10,1,3,1,3,3,2,1,2,32,...
   2b: 277 16; 1,1,1,4,10,1,7,2,2,3,3,2,2,7,1,10,4,1,1,1,32,...
   3a: 310 17; 1,1,1,1,5,3,1,2,1,3,5,1,1,1,1,34,...
   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,...
   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,...
   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,...
   5a: 367 19; 6,2,1,3,1,1,2,1,12,19,12,1,2,1,1,3,1,2,6,38,...
   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,...
   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,...
   5d: 385 19; 1,1,1,1,1,3,1,2,1,3,1,1,1,1,1,38,...
   5e: 393 19; 1,4,1,2,4,1,1,1,1,12,1,1,1,1,4,2,1,4,1,38,...
   5f: 397 19; 1,12,3,4,9,1,2,1,2,1,1,2,1,2,1,9,4,3,12,1,38,...