Amicable numbers from 1 to 100000000

Internat. J. Math. & Math. Sci. 405
VOL. 18 NO. 2 (1995) 405-410

THE UNITARY AMICABLE PAIRS TO I(P

RUDOLPH M. NAJAR

Department of Mathematics
California State University, Fresno
Fresno, CA 93740-0108

(Received January 13, 1993 and in revised form March 19, 1994)

ABSTRACT: We present an exhaustive list of the 185 unitary amicable pairs whose smaller
number is less than 108 and a new unitary sociable set of four numbers.

KEY WORDS AND PHRASES. Unitary amicable pairs, unitary sociable sets.
1991 AMS SUBJECT CLASSIFICATION CODE. 11A25

INTRODUCTION.
A unitary amicable pair (UAP) is a pair of integers m, n such that

6*(m) m + n t*(n) (1.1)

where t* is the sum of unitary divisors function. In 1971, Hagis [4] published a table of UAP’s,
including the results of an exhaustive computer search for all pairs with smaller number less than 106.
There were thirty-two UAP’s in the table, none of which were simultaneously ordinary amicable pairs;
i.e., which were not square-free. The definitions of unitary amicable pairs and ordinary amicable pairs
overlap on square-free pairs. Hagis did not list Wall’s dissertation [7] which included several hundred
UAP’s. Since Hagis, several writers (see, for example [3] and [5]) have added to the known UAP’s.

UNITARY AMICABLE PAIRS.
The present search for UAP’s was conducted on a NeXT station using Mathematica.. Let

s*(n) t*(n) n. (2.1)

The program searched for numbers k such that

s*(s*(k)) k. (2.2)

As part of a verification of the program, it initially picked out fixed points of the first ten iterates of s*,
thus finding the first four unitary perfect numbers.
The search agreed with Hagis’s in finding the nineteen UAP’s less than 106 and the additional
four, pairs number 20, 24, 25, and 111 of this list, less than 108 that Hagis listed. The table follows.

406 R.M. NAJAR

Square free UAP’s are designated by an asterisk. The square-free UAP’s agree with the ordinary AP’s
found in te Riele [6]. Since Wall’s listing [7] was not in numerical order, this list was not compared
against his. Total running time is estimated to have exceeded I(X) hours.

TABLE OF UNITARY AMICABLE PAIRS TO l08

114 2.3.19; 126 2.3(2).7
2 1140 2(2).3.5.19; 1260 2(2).3(2).5.7
3 18018 2.3(2).7.11.13; 22302 2.3(3).7.59
4 32130-- 2.3(3).5.7.17; 40446 2.3(3).7.107
5 44772 2(2).3.7.13.41; 49308 2(2).3.7.587
6 56430 2.3(3).5.11.19; 64530 2.3(3).5.239
7 67158 2.3(2).7.13.41; 73962 2.3(2).7.587
8 *142310 2.5.7.19.107; 168730 2.5.47.359
9. 180180 2(2).3(2).5.7.11.13; 223020 2(2).3(3).5.7.59
10. 197340 2(2).3.5.11.13.23; 286500 2(2).3.5(3).191
11. 241110 2.3(3).5.19.47; 242730 2.3(3).5.29.31
12. 296010 2.3(2).5.11.13.23; 429750 2.3(2).5(3).191
13. 308220 2(2).3.5.11.467; 365700 2(2).3.5(2).23.53
14. 462330 2.3(2).5.11.467; 548550 2.3(2).5(2).23.53
15. 591030 2.3(3).5.11.199; 618570 2.3(3).5.29.79
16. 669900 2(2).3.5(2).7.11.29; 827700 2(2).3.5(2).31.89
17. 671580 2(2).3(2).5.7.13.41; 739620 2(2).3(2).5.7.587
18. 785148 2(2).3.7.13.719; 827652 2(2).3.7.59.167
19. 815100 2(2).3.5(2).11.13.19; 932100 2(2).3.5(2).13.239
20. 1004850 2.3(2).5(2).7.11.29; 1241550 2.3(2).5(2).31.89
21. *1077890 2.5.11.41.239; 1099390 2.5.17.29.223
22. 1080150 2.3.5(2).19.379; 1291050 2.3(2).5(2).19.151
23. * 1156870 2.5.11.13.809; 1292570 2.5.19.6803
24. 1177722 2.3(2).7.13.719; 1241478 2.3(2).7.59.167
25. 1222650 2.3(2).5(2).11.13.19; 1398150 2.3(2).5(2).13.239
26. 1281540 2(2).3.5.13.31.53; 1621500 2(2).3.5(3).23.47
27. 1475810 2.5.7.29.727; 1669150 2.5(2).7.19.251
28. * 1511930 2.5.7.21599; 1598470 2.5.19.47.179
29. 1571388 2(2).3.7.13.1439; 1654212 2(2).3.7.47.419
30. 1610700 2(2).3.5(2).7.13.59; 1883700 2(2).3(2).5(2).7.13.23
31. *1669910 2.5.11.17.19.47; 2062570 2.5.239.863
32. 1707720 2(3).3.5.7.19.107; 2024760 2(3).3.5.47.359
33. 1908420 2(2).3.5.17.1871; 2135100 2(2).3.5(2).11.647
34. 1922310 2.3(2).5.13.31.53; 2432250 2.3(2).5(3).23.47
35. 1997520 2(4).3.5.7.29.41; 2115120 2(4).3.5.7.1259
36. *2236570 2.5.7.89.359; 2429030 2.5.23.59.179
37. 2357082 2.3(2).7.13.1439; 2481318 2.3(2).7.47.419
38. *2728726 2.7.11.13.29.47; 3077354 2.7.19.23.503
39. 2862630- 2.3(3).5.17.1871; 3202650 2.3(2).5(2).11.647
40. 3406116 2(2).3.7.23.41.43; 3690204 2(2).3.7.197.223
41. 3482700 2(2).3.5(2).13.19.47; 3506100 2(2).3.5(2).13.29.31
42. 3951990 2.3(4).5.7.17.41; 49748:8 2.3(4).7.41.107

UNITARY AMICABLE PAIRS 407

43. 4198236 2(2).3.7.23.41.53; 4510884 2(2).3.7.83.647
44. "4246130 2.5.7.60659; 4488910 2.5.23.29.673
45. 4439890 2.5.7(2).13.17.41; 5085710 2.5.7(2).97.107
46. *4532710 2.5.7.13.17.293; 6135962 2.7.71.6173
47. 4918550 2.5(2).7.13.23.47; 5145322 2.7.13.17.1663
48. 5073570 2.3(3).5.19.23.43; 5570910 2.3(3).5.47.43
49. 5109174 2.3(2).7.23.41.43; 5535306 2.3(2).7.197.223
50. *5123090 2.5.7.163.449; 5504110- 2.5.19.59.491
51. 5191290 2.3(4).5.13.17.29; 5967270 2.3(4).5.53.139
52. 5224050 2.3(2).5(2).13.19.47; 5259150 2.3(2).5(2).13.29.31
53. 5341620 2(2).3.5.127.701; 5441100 2(2).3.5(2).7.2591
54. *5385310- 2.5.7.107.719; 5812130 2.5.17.179.191
55. 5431188 2(2).3.7.19.41.83; 5858412 2(2).3.7.97.719
56. 6297354 2.3(2).7.23.41.53; 6766326 2.3(2).7.83.647
57. 6381732 2(2).3.7.17.41.109; 6923868 2(2).3.7.139.593
58. 6433476 2(2).3.7.19.29.139; 7006524 2(2).3.7.239.349
59. 6940890 2.3(4).5.11.19.41; 7937190 2.3(4).5.41.239
60. *6993610 2.5.13.23.2339; 7158710-- 2.5.13.53.1039
61. 7051590 2.3(3).5.7(2).13.41; 7766010 2.3(3).5.7(2).587
62. *7288930 2.5.11.23.43.67; 8221598 2.7.11.197.271
63. 7929012 2(2).3.7.13.53.137; 8763468 2(2).3.7.104327
64. 8012430 2.3(2).5.127.701; 8161650 2.3(2).5(2).7.2591
65. 8095620 2(2).3.5.13.97.107; 9685500 2(2).3.5(3).11.587
66. 8146782 2.3(2).7.19.41.83; 8787618 2.3(2).7.97.719
67. 8194230 2.3(3).5.11.31.89; 9224010 2.3(3).5.127.269
68. 8235810 2.3(3).5.11.47.59; 9182430 2.3(3).5.71.479
69. 8537100 2(2).3.5(2).11.13.199; 8934900 2(2).3.5(2). 13.29.79
70. *8619765 3.5.7.11.17.439; 9627915 3.5.11.23.43.59
71. *8754130 2.5.7.11.11369; 10893230 2.5.757.1439
72. *8826070 2.5.11.19.41.103; 10043690 2.5.31.179.181
73. 9057300 2(2).3.5(2).7.19.227; 9912300- 2(2).3.5(2).19.37.47
74. *9478910 2.5.7.19.7127; 11049730 2.5.71.79.197
75. 9572598 2.3(2).7.17.41.109; 10385802 2.3(2).7.139.593
76. 9650214 2.3(2).7.19.29.139; 10509786 2.3(2).7.239.349
77. 9680310 2.3(4).5.17.19.37; 10511370 2.3(4).5.19.683
78. 9701076 2(2).3.7.11.10499; 10458924 2(2).3.7.89.1399
79.* 10254970 2.5.11.53.1759; 10273670 2.5.11.59.1583
80. 11777220 2(2).3(2).5.7.13.719; 12414780 2(2).3(2).5.7.59.167
81. 11893518 2.3(2).7.13.53.137; 13145202 2.3(2).7.104327
82. 12143430 2.3(2).5.13.97.107; 14528250 2.3(2).5(3).11.587
83. 12805650 2.3(2).5(2).11.13.199; 13402350 2.3(2).5(2).13.29.79
84. 12934680 2(3).3.5.11.41.239; 13192680 2(3).3.5.17.29.223
85. 13321490 2.5.7.13.14639; 16192750 2.5(3).7.19.487
86. 13585950 2.3(2).5(2).7.19.227; 14868450 2.3(2).5(2). 19.37.47
87. 13882440 2(3).3.5.11.13.809; 15510840 2(3).3.5.19.6803
88. 14257020 2(2).3.5.127.1871; 14496900 2(2).3.5(2).11.23.191
89. *14426230 2.5.7.13.83.191; 18087818 2.7.31.71.587
90. 14551614 2.3(2).7.11.10499; 15688386 2.3(2).7.89.1399

408 R.M. NAJAR

91 14634270 2.3(4).5.7.29.89; 17247330 2.3(4).5.107.199
92. 16045722 2.3(3).7.11.17.227; 17048934 2.3(3).7.23.37.53
93. 16326090 2.3(3).5.11.23.239; 18510390 2.3(3).5.179.383
94. *17041010 2.5.7.31.7853; 19150222 2.7.13.43.2447
95. * 17257695 3.5.7.13.47.269; 17578785 3.5.7.23.29.251
96. 17278548 2(2).3.7.29.41.173; 17799852 2(2).3.7.29.7307
97. 17468730 2.3(3).5.23.29.97; 18093510 2.3(3).5.19.3527
98. 17709720 2(3).3.5.7.29.727; 20029800 2(3).3.5(2).7.19.251
99. 18143160 2(3).3.5.7.21599; 19181640 2(3).3.5.19.47.179
100. 20038920 2(3).3.5.11.17.19.47; 24750840 2(3).3.5.239.863
101. 21385530 2.3(2).5.127.1871; 21745350 2.3(2).5(2).11.23.191
102. *21448630 2.5.7.131.2339; 23030090 2.5.19.53.2287
103. 21705684 2(2).3.7.11.13(2).139; 23990316 2(2).3.7.285599
104. 23570820 2(2).3(2).5.7.13.1439; 24813180 2(2).3(2).5.7.47.419
105. 24542700 2(2).3.5(2).7.13.29.31; 31367700 2(2).3(2).5(2).7.13.383
106. 25425876 2(2).3.7.19.89.179; 26414124 2(2).3.11.17.79.149
107. 25917822 2.3(2).7.29.41.173; 26699778 2.3(2).7.29.7307
108. 26355084 2(2).3.7.29.31.349; 27404916 2(2).3.7(2).11.19.223
109. 26791830 2.3(3).5.13.17.449; 30361770 2.3(3).5.139.809
110. 26838840 2(3).3.5.7.89.359; 29148360- 2(3).3.5.23.59.179
111. 27287260 2(2).5.7.11.13.29.47; 30773540 2(2).5.7.19.23.503
112. 29408130 2.3(3).5.17.43.149; 30467070 2.3(3).5.19.5939
113. 29656530 2.3(4).5.19.41.47; 29855790 2.3(4).5.29.31.41
114. *30724694 2.7.11.13.103.149; 32174506 2.7.13.17.10399
115. 31035550 2.5(2).7.13.19.359; 31863650 2.5(2).7.13.47.149
116. 32558526 2.3(2).7.11.13(2).139; 35985474 2.3(2).7.285599
117. 32744712 2(3).3.7.11.13.29.47; 36928248 2(3).3.7.19.23.503
118. *34256222 2.7.11.13.71.241; 35997346 2.7.11.23.10163
119. *35361326 2.7.11.13.17.1039; 40117714 2.7.13.53.4159
120. *37784810 2.5.7.539783; 39944086 2.7.13.41.53.101
121. 38138310 2.3(3).5.7.17.1187; 48081978 2.3(3).7.131.971
122. 38138814 2.3(2).7.19.89.179; 39621186 2.3(2).11.17.79.149
123. 39532626 2.3(2).7.29.31.349; 41107374 2.3(2).7(2).11.19.223
124. 40880532 2(2).3.7.11.151.293; 44920428 2(2).3.7.607.881
125. 42740880 2(4).3.5.7.13.19.103; 52306800 2(4).3.5(2).7.13.479
126. 43181292 2(2).3.7.11.17.2749; 51858708 2(2).3.11.131.2999
127. 48339228 2(2).3.7.17.33851; 49154532 2(2).3.7.53.61.181
128. 49117590 2.3(5).5.17.29.41; 50492970 2.3(5).5.11.1889
129. 50953560 2(3).3.5.7.60659; 53866920 2(3).3.5.23.29.673
130. 51091740- 2(2).3(2).5.7.23.41.43; 55353060 2(2).3(2).5.7.197.223
131. 53278680 2(3).3.5.7(2).13.17.41; 61028520 2(3).3.5.7(2).97.107
132. 54392520- 2(3).3.5.7.13.17.293; 73631544 2(3).3.7.71.6173
133. 58062480 2(4).3.5.7.17.19.107; 68841840- 2(4).3.5.17.47.359
134. 59022600 2(3).3.5(2).7.13.23.47; 61743864 2(3).3.7.13.17.1663
135. 59037132 2(2).3.7.11.181.353; 64664628 2(2).3.7.251.3067
136. 59604468 2(2).3.7.11.251.257; 65226252 2(2).3.7(3).13.23.53
137. 60477900 2(2).3.5(2).7.31.929; 63323700 2(2).3.5(2).11.31.619
138. 61320798 2.3(2).7.11.151.293; 67380642 2.3(2).7.607.881

UNITARY AMICABLE PAIRS 409

139. 61477080 2(3).3.5.7.163.449; 66049320 2(3).3.5.19.59.491
140. 62973540 2(2).3(2).5.7.23.41.53; 67663260 2(2).3(2).5.7.83.647
141. 63022806 2.3(3).7.11.23.659; 64710954 2.3(3).7.11.79.197
142. 63813036 2(2).3.7.17.44687; 64888404 2(2).3.7.41.83.227
143. 64623720 2(3).3.5.7.107.719; 69745560 2(3).3.5.17.179.191
144. 64771938 2.3(2).7.11.17.2749; 77788062 2.3(2). 11.31.299
145. 65156700 2(2).3.5(2).7.19.23.71; 78612900 2(2).3.5(2).31.79.107
146. *66595130 2.5.7.31.30689; 74824390 2.5.31.59.4091
147. 69405490 2.5.7.11.23.3919; 93164750 2.5(3).7.139.383
148. 69662970-- 2.3(3).5.13.89.223; 72585990 2.3(3).5.41.79.83
149. 70370412 2(2).3.7.17.49279; 71555988 2(2).3.7.41.79.263
150. 71151210 2.3(4).5.13.29.233; 73910070 2.3(4).5.13.7019
151. *71241830 2.5.11.19.89.383; 78057370 2.5.17.359.1279
152. 71620500 2(2).3.5(3).7.19.359; 73531500- 2(2).3.5(3).7.47.149
153. 72508842 2.3(2).7.17.33851; 73731798 2.3(2).7.53.61.181
154. 72696690 2.3(4).5.11.41.199; 76084110 2.3(4).5.29.41.79
155. 73284900 2(2).3.5(2).13.19.23.43; 80468700 2(2).3.5(2).13.47.439
156. 75139680 2(5).3.5.7.11.19.107; 89089440 2(5).3.5.11.47.359
157. 75729654 2.3(4).7.11.13.467; 79002378 2.3(4).7.13.23.233
158. 76487964 2(2).3.7.17.29.1847; 83179236 2(2).3.7.131.7559
159. *78447010 2.5.17.19.149.163; 80960990 2.5.11.491.1499
160. 79509870 2.3(3).5.11.19.1409; 91043730 2.3(3).5.449.751
161. 81467820 2(2).3(2).5.7.19.41.83; 87876180 2(2).3(2).5.7.97.719
162. 83093388 2(2).3.7.23.41.1049; 86250612 2(2).3.7.83.89.139
163. 83846532 2(2).3.7.11.103.881; 92271228 2(2).3.7.701.1567
164. 83923320 2(3).3.5.13.23.2339; 85904520 2(3).3.5.1"3.53.1039
165. 84650130-- 2.3(3).5.19.29.569; 87717870 2.3(3).5.19.17099
166. 85305948 2(2).3.7.13.191.409; 91026852 2(2).3.7.59.18367
167. 86075730 2.3(3).5.13.137.179; 89195310 2.3(3).5.19.17387
168. 86490978 2.3.7(2).37.7951; 94814622 2.3(2).7.283.2659
169. 87467160 2(3).3.5.11.23.43.67; 98659176 2(3).3.7.11.197.271
170. *87998470 2.5.7.29.67.647; 102358010 2.5.47.89.2447
171. 88555698 2.3(2).7.11.181.353; 96996942 2.3(2).7.251.3067
172. 89406702 2.3(2).7.11.251.257; 97839378 2.3(2).7(3).13.23.53
173. 90062700 2(2).3.5(2).7.13.3299; 102129300 2(2).3(2).5(2).7.13.29.43
174. 90716850 2.3(2).5(2).7.31.929; 94985550 2.3(2).5(2).11.31.619;
175. 91250124 2(2).3.7.29.47.797; 92609076 2(2).3.7.37.83.359
176. 91885920 2(5).3.5.7.23.29.41; 99714720 2(5).3.5.7.59.503
177. 94283700 2(2).3.5(2).7.17.19.139; 115380300 2(2).3.5(2).7(2).47.167
178. 94327860 2(2).3.5.11.131.1091; 113239500 2(2).3.5(3).11.6863
179. 95719554 2.3(2).7.17.44687; 97332606 2.3(2).7.41.83.227
180. 95725980 2(2).3(2).5.7.17.41.109; 103858020 2(2).3(2).5.7.139.593
181. *95791430 2.5.7.17.101.797; 115187002 2.7.17.113.4283
182. 96502140 2(2).3(2).5.7.19.29.139; 105097860 2(2).3(2).5.7.239.349
183. 97292058 2.3.43.439.859; 102503142 2.3(2).7.43.18919
184. 97735050 2.3(2).5(2).7.19.23.71; 117919350 2.3(2).5(2).31.79.107
185. 98324226 2.3(3).7.11.13.17.107; 121145598 2.3(3).7.53.6047

410 R.M. NAJAR

Hagis posed five questions of which two bear comment in terms of the UAP’s found.
Question 3. Is every odd pair of unitary amicable numbers simultaneously amicable?
Only two pairs listed here, numbers 70 and 95, are odd and both are amicable. The evidence
suggests that the answer is "Yes."
Question 4. If rn 2aM and n 2bN, where MN is odd, is it always the case that a b?
The condition holds for all pairs listed here. Again, the evidence, meager as it is, suggests "Yes."
3. NEW UNITARY SOCIABLE SETS.
Flammenkamp [2] discovered eleven new sets of sociable numbers. Eight sets have four
members; two, eight; and one, nine. We may use

23/(23) 210.3.5.7.41/* (210.3.5.7.41) (3.1)

on Flammenkamp’s fifth set to produce a new four element set of unitary sociable numbers.

58682 84023 96160 2(10).3.5.7.11.19.41.1722307
2. 82978 97587 55840 2(10).3.5.7.17.41.71.343891;
3. 95405 61761 43360 2(10).3.5.7.31.41.13567133;
4. 73443 99131 49440 2(10).3.5.7.41.47.8371211.

The remaining sociable sets in [2] and those listed by Cohen 1] are not amenable to this type of
conversion since a necessary condition of relative primeness fails.

ACKNOWLEDGEMENT. The author wishes to note the assistance of a colleague who prefers to
remain unnamed.

REFERENCES

1. COHEN, H. On Amicable and Sociable Numbers, Mtth. Comp. 2_.4 (1970), 423-429.
2. FLAMMENKAMP, A. New Sociable Numbers, Math. Comp. 56 (1991), 871-873.
3. GARCIA, M. New Unitary Amicable Couples, J. Recreat. Math. 17 (1)(1984-85), 32-35.
4. HAGIS, JR., P. Unitary Amicable Numbers, Math. Comp. 25 (1971), 915-918.
5. MCCLUNG, O. W. Generators of Unitary Amicable Numbers, Fibonaci Quart.
23 (1985), 158-165.
6. TE RIELE, H.J.J. Computation of all the amicable pairs below 1010, Math Comp. 4..Q7
(1986), 361-368; $9-$40.
7. WALL, C.R. Topics Related to the Sum of Unitary Divisors of an Integer,
Dissertation, U. of Tenn. 1970.

Mathematical Problems in Engineering

Special Issue on
Time-Dependent Billiards
Call for Papers
This subject has been extensively studied in the past years Guest Editors
for one-, two-, and three-dimensional space. Additionally,
Edson Denis Leonel, Department of Statistics, Applied
such dynamical systems can exhibit a very important and still
Mathematics and Computing, Institute of Geosciences and
unexplained phenomenon, called as the Fermi acceleration
Exact Sciences, State University of São Paulo at Rio Claro,
phenomenon. Basically, the phenomenon of Fermi accelera-
Avenida 24A, 1515 Bela Vista, 13506-700 Rio Claro, SP,
tion (FA) is a process in which a classical particle can acquire
Brazil; edleonel@rc.unesp.br
unbounded energy from collisions with a heavy moving wall.
This phenomenon was originally proposed by Enrico Fermi Alexander Loskutov, Physics Faculty, Moscow State
in 1949 as a possible explanation of the origin of the large University, Vorob’evy Gory, Moscow 119992, Russia;
energies of the cosmic particles. His original model was loskutov@chaos.phys.msu.ru
then modified and considered under different approaches
and using many versions. Moreover, applications of FA
have been of a large broad interest in many different fields
of science including plasma physics, astrophysics, atomic
physics, optics, and time-dependent billiard problems and
they are useful for controlling chaos in Engineering and
dynamical systems exhibiting chaos (both conservative and
dissipative chaos).
We intend to publish in this special issue papers reporting
research on time-dependent billiards. The topic includes
both conservative and dissipative dynamics. Papers dis-
cussing dynamical properties, statistical and mathematical
results, stability investigation of the phase space structure,
the phenomenon of Fermi acceleration, conditions for
having suppression of Fermi acceleration, and computational
and numerical methods for exploring these structures and
applications are welcome.
To be acceptable for publication in the special issue of
Mathematical Problems in Engineering, papers must make
significant, original, and correct contributions to one or
more of the topics above mentioned. Mathematical papers
regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in
Engineering manuscript format described at http://www
.hindawi.com/journals/mpe/. Prospective authors should
submit an electronic copy of their complete manuscript
through the journal Manuscript Tracking System at http://
mts.hindawi.com/ according to the following timetable:

Manuscript Due March 1, 2009
First Round of Reviews June 1, 2009
Publication Date September 1, 2009

Hindawi Publishing Corporation
http://www.hindawi.com

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