appenda1.htm

Appendix A1

The following tables were built up using both UltraBASIC on DOS, "C" programs on ICL Unix boxes and latterly Meyrignac's Euler2000 program for Windows. Asterisks indicate that the given representation might not be best possible. An empty box in a representation means that there are zero elements of the base involved.

Powers of 4

power	no. of 3's	no. of 2's	no. of 1's	total	predicted
2		4		4	4
3	2	1	2	5	5
4		16		16	16
5	3	9	7	19	19
6	5	7	3	15	15
7	7	8	51	66	66
8	9	25	87	121	121
9	13	12	121	146	146
10	15	159	25	199	199
11	22	145	110	277	277
12	28	463	420	911	911
13	37	991	641	1669	2247
14	43	3831	685	4559	4559
15	66	3867	106	4039	4039
16	88	7734	424	8246	11136

Powers of 5

power	no. of 4's	no. of 3's	no. of 2's	no. of 1's	total	predicted
2	1	1			2	2
3		4	2	1	7	7
4	2	1	2		5	5
5	2	4	3	9	18	18
6	2	10	2	15	29	29
7	4	4	30	1	39	39
8	1	49	14	16	80	85
9	7	6		19	32	32
10	9	4	90	85	188	188
11	9	62	47	19	137	850
12	7	238	53	67	365	820
13	4	595	445	44	1088	1088
14		1272	1195	177	2644	4130
15	15	1003	595	84	1697	4294
16	14	2146	1222	223	3605	8110

Powers of 6

power	no. 5's	no. 4's	no. 3's	no. 2's	no. 1's	total	predicted
2		2		1		3	3
3	1	1	1			3	3
4		5		1		6	6
5	1	4	2	2	5	14	14
6	2	3	4	3	10	22	22
7		17		11		28	36
8	4	1	7	22	21	55	55
9	2	19	60	19	2	102	102
10	4	20	6	76	38	144	144
11	3	50	37	21	34	145	291
12		121	276	15	44	456	547
13		180	615	72	27	894	1055
14	1	246	1301	176	42	1766	3855
15	2	377	303	44	65	791	2905
16	5	479.	20	308	39	851	3503

Powers of 7

power	no. 6's	no. 5's	no. 4's	no. 3's	no. 2's	no. 1's	total	predicted
2	1			1	1		3	3
3	1	1				2	4	4
4	1		4	1			6	6
5	1	2	2	3		4	12	12
6	2	1	2		8	8	21	21
7	1	5	9	2	9		26	26
8		14	3	15	4	4	40	40
9	1	14	9	29	4	10	67	67
10		27	17	16	32	30	122	122
11	5	3	4		39		51	51
12	5	10	29	55	49	48	196	196
13		49	551	61	39	27	727	1046
14	2	45	919	30	53	46	1095	3586
15	8	17	435	14	129		603	2509
16	10	32	29	336	19	17	443	1412

Powers of 8

power	no. 7's	no. 6's	no. 5's	no. 4's	no. 3's	no. 2's	no. 1's	total	predicted
2				4				4	4
3		2		1		2		5	5
4				16				16	18
5		3	3			2	1	9	9
6		4	4	3	1		3	15	15
7	1	2	8	5	3	2		21	21
8	1		28	1	1	11	2	44	47
9		12	6	5	13		27	63	72
10	1	11	8	45	14	1	9	89	133
11	4	1	5	16	37	13	12	88	88
12	2	18	6	22	39	37	33	157	404
13			446	78	55	5	21	605	999
14		13	549	105	62	2	5	736	3206
15	2	43	179	8	4	28	79	343	1886
16	6	27	32	238	56	7	162	528	1140

Powers of 9

power	no. 8's	no. 7's	no. 6's	no. 5's	no. 4's	no. 3's	no. 2's	no. 1's	total	predicted
2			2			1			3	3
3	1		1					1	3	3
4			5			1			6	6
5	1	1		3			3	3	11	11
6		4		2	7	1	3	2	19	22
7	1	2	3		12	1	1		20	25
8	1	4		8		13		8	34	34
9	1	4	7	10	6	7	5	6	46	46
10	2	2	9	23	5	5	7	3	56	67
11	2	6	4	17	13	8	10	10	70	92
12	1	12	21	7	11	16	36	14	118	157
13		6	119	329	70		28	2	554	884
14		3	244	281	23	1	31	4	587	3043
15	3	18	23	131	65	4	20	8	272	389
16	3	23	86	10	21	147	38	13	341	579

Powers of 10

power	no. 9's	no. 8's	no. 7's	no. 6's	no. 5's	no. 4's	no. 3's	no. 2's	no. 1's	total	predict
2		1		1						2	2
3			2		2	1				5	5
4		2		1		2				5	5
5		1	4						4	9	9
6	1	1		4	1	1		1	6	15	15
7	1		5	3	3	1	4		1	18	23
8		3	8	2		1	19	2	5	40	41
9		7		6				19		32	44
10	1	4	7	3	4	19	9	6	3	56	56
11	1	6	4	23	16	10	14	10	1	85	103
12	2		30	9	1	4		5	15	66	79
13		6	17	385	20	23	17	9	2	479	878
14		5	80	300	39	17	15	8	26	490	2907
15	3	3	56	21	33	27	6	31	30	210	519
16		30	46	9	9	63	47	94	26	324	732

Powers of 11

power	10's	9's	8's	7's	6's	5's	4's	3's	2's	1's	total	predict
2				1	2						3	3
3			1	2		1			1		5	5
4	1		1				2		2	1	7	7
5	1	1					1	4		6	13	13
6		2	1	2	4	1	2	1	1	3	17	17
7	1	1	1	2	3	1	2	4	4	3	22	22
8	1	1	2	5	1	18	3	4		5	40	44
9		4	5	3	1	3		9	4	8	37	53
10		6	4	1	6	6	17	1		3	44	44
11	2	2	2		13	13	4	10	3	26	75	105
12		10		22	4	3	11	1	15	13	79	92
13		2		294	70	32	4	20	3	3	428	876
14			53	212	33	46	12	16	10	7	389	2339
15	2	6	22	35	2	24	7	16	13	32	159	318
16	2	10	20	52	20	36		16	4	79	239	647

Powers of 12

power	11's	10's	9's	8's	7's	6's	5's	4's	3's	2's	1's	total	predict
2				2				1				3	3
3		1		1		1						3	3
4				5				1				6	16
5	1		1		1	1	1	1				6	6
6			5			7			3			15	15
7		1	2	5	7				7			22	24
8	2						3	1	4	1	23	34	34
9	1	1	1	7	11	1	10	5	4		1	42	50
10		3	8	2	6	2	6	1	3	2	1	34	45
11		4	10	2	5	3	21	4	4	2	6	61	64
12		7	4	8	16	6	8	9	2	4	2	66	66
13			7	133	160	41	36	14	4		1	396	852
14		1	16	179	41	36	1	19		6	14	313	2334
15	1	6	25	2	2	4	23	45	19	11	2	140	276
16	1	13	1	22	26	7	33	51	22	1	45	222	290

Powers of 13

power	12's	11's	10's	9's	8's	7's	6's	5's	4's	3's	2's	1's	total	predict
2	1							1					2	2
3			1	1		1		1					4	4
4	1				1	1	1				2		6	6
5		1	1		2	2	1		3	1		1	12	12
6		1		4	3	1		1	2	2	2		16	16
7			3	5	4		1	2		4	1	2	22	23
8		1	2	8	1	6	1	10		7	1	1	38	38
9	1		5		1	6	6	4		5	5		33	35
10	1	1	4		9		5	3	8		10	5	46	46
11	1		7	9	6	6	9	1	1	4	7	1	52	55
12		5	5	6	13		8	2	6	18	13	2	78	78
13		1		48	265	6	5	8	17				350	820
14		2		128	53	20	38	1	9		5	10	266	2060
15	3			22	11	9	11	9	19	10	2	13	109	153
16	2	2	13	34	37	9	11	8	5	8	4	4	137	324

Powers of 14

power	13's	12's	11's	10's	9's	8's	7's	6's	5's	4's	3's	2's	1's	total	predict
2		1						1		1				3	3
3		1		1								2		4	4
4		1				4		1						6	6
5	1		1						1	2	1	2		8	8
6	1			2	1			3	2			1	4	14	14
7			1	8		2	2		1		3	2	1	20	25
8		1	4	1	1	1	4		14		3	1	6	36	36
9	1	1	1	2	1			14	5	2	1	3	3	34	41
10		2	5		8	7		5	2		8	1	1	39	50
11		4	3	2		2	2	1	1	12	1	9	1	38	38
12			15	6	9	15	3	1	4	9	1	3		66	81
13		1		4	250	14	36	5	4	5	2	1		322	776
14		1	10	39	82	56	11	12		6	1	5		223	1725
15	2	1	1	27	29	18	1	6	10	4	8	10	13	130	202
16	3		1	10	18	5	29	10		17	70	1	7	171	239

Powers of 15

pow	14's	13's	12's	11's	10's	9's	8's	7's	6's	5's	4's	3's	2's	1's	total	pred
2			1			1									2	2
3				2				2				1			5	5
4			2			1			2						5	5
5			2	1			2	2			1	2			10	10
6			2	3					1	3	2	3	1		15	15
7			2	4	2				3	5	1		2	2	21	21
8		2	1	2			1	9	1	6		7		7	36	37
9		2	3		1	1	2	2	1	4	2	2	5	4	29	29
10	1	1	2		1	4		6	1			1	1	5	23	23
11		1	7	2	10		10				3	9	3	1	46	53
12	1		3	14	2	1	1	1		4	9	17	1	12	66	66
13	***	1			133	122	2	20	12	15				1	306	791
14			4	36	102	7	5	6		10		1	5	13	189	1546
15	1		14	15		19	1	4	3	41	2	13		7	120	157
16	1	2	15	2	17	10	21		6	2	11	21	18	28	154	188

Powers of 16

pow	15's	14's	13's	12's	11's	10's	9's	8's	7's	6's	5's	4's	3's	2's	1's	total	pred
2								4								4	5
3		1				1			1					1	1	5	5
4								16								16	16
5				3		3						2		1		9	11
6		2					2	1	3		2	2	1			13	13
7	1			2	1			3			1	3	3			14	14
8			1	1	13		5		8		3	1			2	34	36
9	1	1		1		4		3	1		6	2	2	2	6	29	29
10	1	1		3			13	1	5		4	1	1	4		34	46
11		1	7	1		1	4	3		8	6		2	2	5	40	40
12	1	1	1	7	2	1	7		3	13	1		16		1	54	66
13	***	1			73	97	84	9	13		2	5	1	3	1	289	763
14			7	10	75	29	11	4	8	11	2	1	1	7		166	919
15		5	3	14	1	1	1	6	48	6	3	4	20	6	2	120	127
16	2		5	7	12	12	5	28	3	4	2	13	28	9	7	137	171

Powers of 17

pw	16's	15's	14's	13's	12's	11's	10's	9's	8's	7's	6's	5's	4's	3's	2's	1's	total	pred
2		1							1								2	2
3					1	1	1	1				1					5	5
4			2						1		2					1	6	6
5		1			1	1	2		1	1			1				8	8
6	1			1			1	2		4					1	2	12	12
7			2	2		2	3	1				3	1	3			17	19
8	1	1							7				7		1		17	37
9		1	2	2	1	4	1	5		2		2		7		2	29	29
10			2	3	16		3		3			1		5	1		34	36
11		2	2	2	4	7	1	7					12	3	2	4	46	46
12		3	3		1	3	1	14			7	3	7		1	10	53	79
13	***	3		9	9	8	6	15	3	3	68	30	46	354	165	15	734	734
14		1	4	12	12	81	12	5	1	4	1	4	3	5	2	1	148	1039
15	1	3	1		11	16	3	3	8	8	3	1	2	6	9	2	77	127
16	1		12	3	6	16	19	21	32	10	1	6		8		1	136	153

Powers of 18

pw	17's	16's	15's	14's	13's	12's	11's	10's	9's	8's	7's	6's	5's	4's	3's	2's	1's	total	pred
2						2						1						3	3
3			1			1			1									3	3
4						5						1						6	6
5	1					1	1				3		3			1		10	10
6	1			1				1	2		2		3			1		11	11
7		1	1	1	1					2		2			4	2		14	14
8			2		6		4	1			7		5		8	1		34	35
9		1		3	5	1	4					4	1	1	4	1	2	27	29
10		1	1	4	3	5		1	1		2		7	4		3	2	34	36
11	1		1	4	1	1	8	2	3	4		6	3	2	3			39	45
12		3	1	1	4	3	1	1	4	11			6	2		7		44	44
13	***	4			7	3	6	11	16	14	8	88	25	42	311	153	9	697	697
14		2	3	7		28	75	9		1				4	1		1	131	547
15	1	1	4	4	2	10	22	8	4	7	7	7	9	1	1	5	2	95	96
16	1	1	7	3	1	3	13			4	10		16	23	7	6	9	104	139

Powers of 19

pw	18s	17s	16s	15s	14s	13s	12s	11s	10s	9's	8's	7's	6's	5's	4's	3's	2's	1's	total	pred
2				1					1				1						3	3
3	1								1							1			3	3
4	1						1				1				2			1	6	6
5			2					1	2			1			1		2	1	10	10
6		1				4	1				1	3						1	11	11
7		1		2	1		1						2			4	3		14	14
8	1				2	1	4		3		10		5		3				29	36
9		2			2	1	6		2	1	1	2	2	2	1		1	3	26	26
10	1			2	2		9	10	1		2			1	1	2		2	33	35
11		2	2		2	2		3		6	4			1	3	3	8	2	38	38
12	1	1		1	5	2	1		3	7	2	1	2	2		2	11	1	42	42
13																				723
14																				577
15		1	9	4			9	11	4		3	11	11	3		10	6	1	93	120
16	2		2	1		2	2	8	3	1		10	23	33	6	1		9	103	103

Powers of 20

pw	19s	18s	17s	16s	15s	14s	13s	12s	11s	10s	9's	8's	7's	6's	5's	4's	3's	2's	1's	total	pred
2				1				1												2	2
3			1			1							1							3	3
4				2				1				2								5	5
5				1		4												4		9	9
6	1					2				1		2	3		1		3			13	13
7			1	2		1	3		2						2			6		17	19
8					4	1	15	1	5		3		1				1		3	34	36
9	1		1			1	1	7	1		2		5		3			2		24	24
10		2	1		1	1	1	1			4	1	6	11	1		2	2		34	34
11		1	1	4		7	3	2	2		1	9		6		1	3	2		42	42
12			5	3	1	1	6	1	1			2	2	7		14		2	1	46	66
13																					649
14																					505
15		4		2	5	6	5	5	2	11	1	4	10	7	1	4		3	10	80	114
16			10	7	3	5	13		8		4	1	4	14	8	2	14	3		96	122

URL : www.glasgowg43.freeserve.co.uk/appenda1.htm

Last updated : 07/06/2000