ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS
RICHARD J. MATHAR
Abstract. Aliquot sequences are illustrated by assembling members of prime families into a single graph, with arrows pointing to the next member of aliquot sequence.
1. Intro An aliquot sequence is an iterative mapping of positive integers n → σ(n) − n, from n to the sum of its divisors that are less than n [5,1,8,7][6, Sec. B5]. These connections are tabulated in entry A001065 of the OEIS www.oeis.org [9], and can be investigated by the database in www.factordb.com. The typical track of such a mapping ends in a prime number, and this prime number is followed by a 1, which is generally not mentioned explicitly. The length of the track is entry A098007 [9]. In other cases, the track may enter a cycle, see A098007 in the OEIS [9], even a cycle of length one. In any case, we may build families of the integers having common parts of their tracks (i.e., having the same termination, A115350 in the OEIS [9]). The graphs in section2 show some of these families, whereas section3 considers the map n → σ∗(n).
2. Aliquot sequence Illustrations The following plots up to Figure 72 show some of these mergers. Where they end in a known prime, that prime is marked with a p. Obviously, each of the plots is only a part of a tree-type structure. Because I generated them by tracing a finite set of initial values, the “early” plots may appear to have more members, but this is an illusion generated from this statistical sampling artifact. In addition, I have not traced the sequences completely in all cases; this means the individual families are not complete up to the maximum number that appears in the individual plot! The plots are created with the dot command of www.graphviz.org.
Date: March 21, 2014. Key words and phrases. Aliquot Sequence, Divisors. 1 2 RICHARD J. MATHAR
Figure 1. Prime family 3 [9, A127163] 4313 3893 3029 3835 247 1673 3603 1205 3115 2735 1273 2937 4351 3199 949 3663 2265 1687 553 1383 2569 87 3039 249 375 3661 2259 1017 531 288 4089 2623 3365 3987 3129 2479 1671 465 4437 2583 1785 2263 105 4257 3687 4455 679 33 4395 2607 1233 561 303 891 2661 1258 207 2509 2401 1222 1393 400 3862 794 1934 970 4453 4454 2674 524 2078 1042 135 3589 1352 1528 372 4036 3034 1985 3193 4415 403 880 2782 1754 4898 889 4541 45 3367 884 2422 3489 259 4097 4927 4623 3518 1762 1905 9 4 3p 15 1757 393 648 522 1167 402 414 144 90 615 3327 4249 3397 759 1113 426 2343 2223 1957 78 342 438 450 123 4699 1417 246 258 270 174 4277 165 1963 66 186 198 102 126 114 72 2037 1099 54 2809 3591 86 326 166 42 1681 46 26 16 4199 52 212 30 841 2135 3599 3551 3431 12 3239 3071 2759 121 3225 2231 1919 4585 1751 1391 4945 1199 2361 2475 791 4237 243 2977 4963 717 429 2947 ALIQUOT SEQUENCE MERGERS 3
Figure 2. Aliquot cycle 6. 4897 4717 4141 4739 3277 2725 143 1441 2387 2541 685 1715 1443 4317 417 2863 909 3337 783 2581 4775 6 25 119 1717 3899 1177 565 2225 1595 675 3999 2173 95 1633 3575 913 4535 3311 4763 445 2195 3693 1235 4 RICHARD J. MATHAR 4911 1641 551 3843 2605 527 3501 1569 2005 3471 2145 407 1105 4475 4569 4573 1887 3657 1405 49 287 1527 849 2505 513 4785 4438 3855 3057 1023 4393 3904 3970 3194 1600 2337 215 735 3349 3232 2042 3650 1024 4078 4715 992 633 8 3538 7p 3935 3543 1333 1185 75 793 2703 3175 2793 1767 2637 4336 3227 4096 3344 3314 1660 469 1868 2315 3042 4095 4641 1408 1288 1592 3423 1825 1652 1708 1764 1547 2824 2486 4629 868 924 812 1618 213 3667 1832 4821 420 364 729 4329 2587 1611 4935 4281 10 308 4847 1431 2889 3255 4247 1845 1995 4031 2831 4045 14 169 2567 2433 1727 2065 815 575 2845 4047 1713 363 4537 4511 1077 3839 38 22 1369 2471 20 361 2299 4645 935 34 867 2589 62 3721 188 148 118 668 388 508 298 152
Figure 3. Prime family 7 [9, A127164]. ALIQUOT SEQUENCE MERGERS 5 4675 2021 4383 1961 1829 4795 1769 3951 1541 3711 91 1349 2763 1241 4315 869 3451 2875 581 4029 1731 1435 4901 589 3509 2597 481 1805 51 21 11p 301 4811 2051 4891 4819 4171 141 3811 3165 18 1923 3379 3069 645 4431 1651 2353 4418 411 195 1707 573 3315 3309 4607 1107 3020 915 3364 2733 3047 3692 2748 3523 4155 1967 2517 289 1939 4299 1573 285 4995 4125 843 1749 3363 1421 3861 4875 4345 2355 1437 1415 3405 4905 3675 483 2859 3245 1075 3393 4083 957 2067 2765 3957 1323 3243 576 2565 1365 2235 861 435 1275 2373 3003 4965 2571 2085 3435 4245 3867 2108 1800 1924 1293 4557 2739 1827 2385
Figure 4. Prime family 11 6 RICHARD J. MATHAR 4976 4696 4796 3940 4376 4427 4444 4628 4124 3100 3844 3107 4192 4804 2780 3968 4595 2651 3776 253 1788 2412 4490 3610 3248 925 3725 2022 2034 2900 845 4195 1566 4559 2755 4343 4223 3959 3503 4401 2159 145 4765 1703 3145 959 695 4655 2185 35 539 2073 475 3269 1891 3605 93 1387 3773 1027 968 267 13p 789 4331 4307 4187 3827 3131 27 133 2987 2507 3981 2147 1893 1331 635 1885 3777 2095 425 2769 1263 1015 3913 69 1147 927 427 2933 2105 4707
Figure 5. Prime family 13 ALIQUOT SEQUENCE MERGERS 7 3647 2891 24 36 529 2615 713 2127 2075 4465 629 1295 1771 3873 55 533 2335 473 17p 329 3487 711 4439 39 3743 3625 3383 3153 217 1055 1425 3177 847 3845 775 2465 455 4873 2245
Figure 6. Prime family 17 8 RICHARD J. MATHAR 1363 4985 1219 2429 1003 77 355 3897 1745 3781 219 2397 1477 2443 357 2301 1059 4685 3829 19p 4433 555 1173 943 885 2643 3857 693 2849 1419 799 2885 65 583 1209 2247 4709 4575 295 3653 3295 3113 4929 665 1983 3237 1467 4681 3339 4599 2277 3097 183 3627 2197 537 1179 297 3679 3807 2001 879 3139 4305 3759 639 4417 2923 4917 765 2283 1843 3765 117 3785 4275 1339 2821 763 4291 339 2317 621 1851 382 100 758 194 238 124
Figure 7. Prime family 19 ALIQUOT SEQUENCE MERGERS 9 1763 1739 2325 1643 1403 4197 85 1139 4563 2757 923 3985 803 2365 4213 395 1945 3515 23p 1045 3831 2853 1281 703 2349 1625 559 1911 3641 57 4863 343 1685 2991 1001 578 3937 159 2641 1922 1057 2257 99 2077 1501
Figure 8. Prime family 23 10 RICHARD J. MATHAR 4531 2109 4947 3247 221 4983 1682 2313 2167 3993 4381 1863 931 209 351 1695 1041 2673 609 4207 3921 3799 3155 1311 3009 2178 3151 2525 3219 1341 1639 31p 161 882 1955 637 819 471 3241 3123 333 1401 2151 3763 969 1215 3403 125 627 4333 2323 3325 1635 1005 1819 3555 2685 3965 1243 32 961 3717 2523 68 58 4489
Figure 9. Prime family 31 ALIQUOT SEQUENCE MERGERS 11 4975 2468 542 1858 4823 1225 374 932 1048 1388 706 356 274 772 1538 296 586 710 84 140 500 592 2228 3118 1678 766 842 1030 3188 2398 1562 386 196 424 556 982 2564 1930 4117 658 494 2390 898 346 452 176 2101 203 410 788 598 985 3335 718 597 2601 1779 1838 362 184 926 430 3421 1390 1844 4198 922 1130 464 466 236 4868 323 4895 1585 4814 340 416 3658 2102 674 4484 37p 1054 3916 3644 299 2878 1442 4777 3272 4961 4394 4316 2120 2740 1396 1852 3815 1465 2746 3056 2896 4319 625 156 4633 1376 2348 1768 2012 1516 1144 3479 96 896 3013 2894 3095 155 3695 2329 2222 4749 1587 3782 1450 745 2170 2375 2438 1340 2690 1180 453 3574 1790 1347 3478 4388 3298 1994 1000 2758 4858 3494 1750 2642 3506 1324 1756 4838 1364 2722 4352 4846 4234 2426 1216 3874 1264 1236 1676
Figure 10. Prime family 37 12 RICHARD J. MATHAR 4630 3722 2743 3493 4946 3051 146 507 225 178 1509 3292 2588 88 92 585 1948 477 4039 2476 76 1864 2608 1468 48 2209 1108 998 502 1646 3662 4168 838 254 1046 322 422 3182 4150 130 122 164 692 64 526 266 214 3026 1834 56 1334 4886 206 3514 584 3152 826 2534 1312 1010 106 110 4055 310 614 3748 4534 2270 4346 398 104 202 848 802 404 3326 817 3010 3275 2818 80 698 352 1102 3945 4142 1666 1412 4330 63 2774 3482 2983 1066 3634 2391 2126 2986 1496 3926 4371 1744 177 2119 2458 1232 230 1310 1332 3806 3939 1773 4169 519 801 1377 2614 2792 369 2907 4852 391 2681 3646 1826 1328 41p 4474 3148 3572 1071 2368 3856 2240 3133 4119 255 3377 1588 4042 319 768 1198 4562 4130 2284 4510 602 1720 454 753 4233 1599 1815 4869 2177 3742 1088 3324 4460 4948 2870 3718 3178 2294 4646 3927 2985 1551 1565 2902 1276 3238 1454 3334 4634 256 1670 1874 228 332 1354 2881 730 2698 1622 1118 2844 4436 1774 1796 1116 1244 940 964 2428 1828 2701 1378 4761 4906 2302 680 1076 580 1154 596 814 890 1186 1342 111 2449 3158 1582 3592 4406 2883 2174 4067 4131 1090 764 2804 1522 2110 1261 2932 2421 1089 3703 2206 1106 3195 4862 4210 3386 1696 1706 748 4718 721 2788 640 836 844 2504 664 2639 856 2252 1668 2258 1132 3394 4516 1700 280 440 554 4959 3991 740 520 224 4090 3290 3992 3508 2638 1322 742 4695 2841 321 2191 4828 3190 4244 3622 3830 670 3254 1630 4322 951 3447 1545 3820 4982 2164 2876 405 3585 2175 2779 3082 4958 792 2456 1548 2829 1203 4936 4334 1478 1965 4370 4658 4270 910 2794 1814 3878 4186 4432 2506 3470 950 3340 3716 2056 3176 2944 1894 3614 3110 4498 2266 2810 2732 3490 2384 2628 4106 4762 1178 3950 1846 2452 4994 3656 3214 1610 4528 3620 4276 3536 4024 3974 1990 3416 1200 2644
Figure 11. Prime family 41 ALIQUOT SEQUENCE MERGERS 13 1766 886 446 1172 292 226 4834 2420 268 208 3166 1586 3031 1348 1018 4921 4899 116 2013 632 1083 1591 2585 4323 448 568 441 963 1917 1159 3521 300 204 495 871 3567 4017 3409 1729 81 511 4353 132 4046 3322 1473 3362 2751 512 1455 3849 4978 3073 4687 2150 1989 4731 1287 4822 4304 1329 153 1807 1942 2942 4521 2103 4066 897 447 3710 2414 1474 4659 3346 705 1557 2068 1964 1480 1940 4551 855 1833 2176 484 4202 536 490 974 2186 1096 2710 1840 2624 4009 684 1136 3345 1078 2919 492 348 231 94 1561 4678 2342 2961 590 4711 1174 3460 3848 4604 4972 1556 4132 3106 4497 2031 1798 3002 3730 681 544 1082 3201 1503 1150 716 1479 3417 4429 516 1239 4203 1881 147 4033 1095 3273 2413 4215 4303 2553 345 3531 2359 2229 4835 3045 2715 747 1653 973 3925 4419 3731 4407 278 142 1977 40 50 74 2915 3465 4023 70 44 3111 1849 663 1353 4053 2727 2155 1822 262 134 82 3021 158 4833 2918 1462 3135 3183 1065 437 1299 4293 914 2625 1246 2367 1665 2205 825 2463 43p 4065 377 2241 1119 1202 604 341 1335 4839 1011 1617 2098 796 1052 460 4382 380 548 2839 3609 4815 572 1138 1886 3154 2964 4876 418 4196 782 472 428 1869 4611 1903 2860 1544 328 302 185 1366 686 260 514 895 2945 154 3507 1424 182 2686 1634 1006 543 358 4849 506 3988 754 2998 1502 387 2653 2038 2708 1022 3453 1149 1155 610 1214 2196 902 3446 4527 3105 2655 2025 2295 1726 662 1808 1318 1972 866 436 334 170 1718 3412 862 4582 2618 2566 1298 3218 434 1612 1524 2606 646 1286 1748 4462 656 1306 2054 2060 574 190 336 4318 1732 2594 1142 4606 3488 3598 2308 1738 4274 4786 3602 3442 1300 1724 958 482 244 1268 4282 2396 1804 2140 2144 2858 1432 770 626 1640 1892 3274 3778 4366 1508 3104 2474 818 412 1240 316 3070 1888 746 2966 1486 920 344 408 672 376 4084 2720 1344 368 3098 1552 4966 304 2168 4486 1912 192 2246 2978 4102 3028 1492 1688 3670 4262 1126 1168 2278 3094 4298 2954 2134 1406 2836 2752 1884 3266 1394 1918 2540 3994 4832 4942 3554 874 3022 2000 2092 1070 852 1580 1780 2680 3440 4744 1164 4166 2482 4178 1784 3868 3308 1514 1576 2086 2488 2192 4594 2300 2908 1472 4070 4138 2072 2188 566 760 1012 1004 1628 1460 1564 1648 3728 3526 2018 988 732 972 4378 1040 3382 4598 860 708 3788 1714 2822 3208 3734 1870 2914 4358 2218 1112 2182 2848 2552 2590 2882 2276 550 2338 1694 1498 1094 624 4546 4924 3700 3824 3566 3616 4672 4726 3008 3074 3088 1786 3934 2968 3512 2834 4238 2926 3524 2650 2372 3448 3032 3628 3260 1400 2320 2728 2668 2432 4568 4012 4856 3548 1280 4264 4556 2052 4850 4514 3396 2554 4082 956 724 1664 1906 1960 3170 4918 2462 620 1234 2006 4006 444 286 272 1636 3418 1712 538 806 904 1016 290 250 2692 136 218 112 468 2026 458 232 4402 2510 3958 622 248 1238 160 2366 172 3058 1982 406 314 108 2990 4642 60 994 370 734 2248 1598 3481 1358 4478 2242 2846 3682 2654 1330 1550 1426 2530 878 1252 2498 2806 2984 638 976 946 2626 1658 442 2768 832 530 2612 712 488 4294 1418 2518 3062 1262 4702 2546 1966 320 3458 986 634 3262 2354 1534 4808 4222 2114 3410 3502 2036 2350 4694 2666 962 1558 3310 4214 1862 4526 928 2578 1292 1228 854 3428 3818 1162 948 4466 2090 2230 4174 1802 560 4250 4052 1742 1114 3046 1526 3464 2734 1370 3794 2158 1256
Figure 12. Prime family 43 14 RICHARD J. MATHAR
4653
1245 2835
459 771 4579 3595 2973 2781
261 2071 3007 4087 725 995 1379 2123 2483 4163 2435
129 205 493
47p
Figure 13. Prime family 47
3213
2547 3895
1145 1589 2453 4853 1085 3101 4829 3305 4613
235 451 667
53p
Figure 14. Prime family 53 ALIQUOT SEQUENCE MERGERS 15 4916 3838 3764 2830 2282 4706 3263 1654 3338 3908 830 1799 518 934 1672 3694 265 3398 2062 4502 682 394 200 470 2938 1382 1928 1702 350 1850 3122 2254 697 59p 3455 1684 1034 1270 694 517 2290 4574 3563 938 1946 2410 1414 3902 2496 4616 3044 4054 980 3014 3998 2030 2002 4552 916 3898 2702 1954 4612 4384 2050 1952 4310 3758 3466 1736 1856 4148 3664 2104 944 1882 4688 4426 2216 2330 3392 2064 588 1008 532 476 252 820 2866 3812 1436 1084 4738 2798 2750 704 2378 4772 3586 1402 2318 4742 2662 1730 3086 4154 776 2374 1190 3814 1910 1546 4570 3674 1616 4990 4010 3226 3064 2696 4076 1978 3316 4412 3422 3956 3784 4136 4504 4622 2816 2494 3632 3436 2584 3842 2936 1776 1466 2314 736 3206 752 1810 3076 2404 1642 824 2446 4022 850 1124 2014 616 2786 1226 1510 3374 2762 1384 3236 2434 1220 1160 3038 2492 2548 1540 3368 3832 2962 1484 756
Figure 15. Prime family 59 16 RICHARD J. MATHAR 138 234 150 222 960 528 312 2088 3762 5598 168 264 456 744 3804 5100 2244 1176
Figure 16. Other branches of prime family 59
2500
2055 2967 2799 2079 2895 3321
1257 1761
2575 3215 4487 1659 2795 3995 4825 162 423 591 1351 3439
649 901 1081 1189 201
71p
Figure 17. Prime family 71 ALIQUOT SEQUENCE MERGERS 17
4778
1068 1404 4030 2392
1452 2516 4538 4034 2648
3638 3686 3854 2272 2020 2332
804 2194 2264 2096 2204 3986
1100 1996
1504 3304
1520 3896
2200 3424
3380 4904
4306
2156
4028 2632
3532 3590 1728 3128
1956 2656 2890 4258 2316 3544 3352 4184 4892
2636 2132 3116 2948 3676
3928 1984 2764
1760 3452 2080 2388
2776 1812 2596 4882 3212
2444 3004 2408 3626
216 2260 2872
384 2528 4810
636 2512 4094 4766
3399 876 2386 2602
1593 1196 1304 1532 2306
807 4879 1156
273 1169 1793 2669 3473 993 3553 3805
98 175 335 671 767 1007 1247 1271
73p
Figure 18. Prime family 73 18 RICHARD J. MATHAR
4191
1296 1953 2115 4865 4137 4059
1795 3883 2455 1375 1629 1855 2199 3099 2493 3351 2943 4359 3955
365 497 737 1037 1121 1457 1517
79p
Figure 19. Prime family 79
4837 3185
699 1603 2899 3059 3125 4355
237 781 1357 1537
83p
Figure 20. Prime family 83
795 4459
501 1141 2041 2869 4309 2045 4025
171 415 1207 1711 1927
89p
Figure 21. Prime family 89 ALIQUOT SEQUENCE MERGERS 19
4624
4893 4035
2595 2445 4461 3525
1581 2157 2613 4085 1491 2427 2535 4149
723 1195 2563 3859 813 1345 2893 4369 1857 3085
245 275 623 1079 1343 1679 1943 2183 2279
97p
Figure 22. Prime family 97
3698 4802
1981 3601
291 979 1411 2059 2419 2491
101p
Figure 23. Prime family 101
2395 3459 4923
485 1157 1577 1817 2117 2201 2501 2537
103p
Figure 24. Prime family 103 20 RICHARD J. MATHAR
4743
4533 2745 3381 4605 3597
1515 1947 2091 2139 2787 4235 1683
933 2149 1125
315 903 1975 2495 3179 3751 3887 4251
309 505 1513 1909 2773
107p
Figure 25. Prime family 107
3075 4011 4587 3825
2219 3443 4043 2545 3505 2133 3189 3429
325 515 707 1067 1691 2291 2627 2747 2867
109p
Figure 26. Prime family 109
4069 2048
327 535 1111 2047 2407 2911 3127
113p
Figure 27. Prime family 113
128 1469 1853 2033 2369 2813 3293 3569 3713 3869 3953
127p
Figure 28. Prime family 127 ALIQUOT SEQUENCE MERGERS 21
738
1267 4867 2611 4771 900
189 381 1921 2461 2929 3649 3901 4189
131p
Figure 29. Prime family 131
4503
1897 4529
279 655 2599 3103 4183 4399
137p
Figure 30. Prime family 137
4555
917 1397 3161 3317 3737 3977 4661 4757
139p
Figure 31. Prime family 139
1507 2227 4387 4747
149p
Figure 32. Prime family 149 22 RICHARD J. MATHAR
3885 4545
3411
1529 1781 2489 2921 4181 4469 4601 4841
151p
Figure 33. Prime family 151
4845
628 3795
478 2253 3117 3565
242 755 1043 2363 2603 3683 4859
157p
Figure 34. Prime family 157 ALIQUOT SEQUENCE MERGERS 23
3879
1737 4971 2775 3159
785 1661 1937 3197 4061
163p
474 282
486 294
894 606 390
906 618
1110 1254 1614 918 630
1626 1242
1638
2730
5334
Figure 35. Prime family 163
489 2533 3973
167p
Figure 36. Prime family 167
1533 4145
835 3427
173p
Figure 37. Prime family 173 24 RICHARD J. MATHAR
3475 4295 4949
865 1837 4321
179p
Figure 38. Prime family 179
1701 3685
1211 2171 2771 3611 4379 4619
181p
Figure 39. Prime family 181
4413 4491 3136
1475 1895 2009 4103
385 1969 2941 3841
191p
Figure 40. Prime family 191
908 3915
688 892 1346 3285
676 1803 2995 1647 2487 4135 4753 4165
605 833 1991 2327 3287 4727
193p
Figure 41. Prime family 193 ALIQUOT SEQUENCE MERGERS 25
3997
579 955 3043 3979 4843
197p
Figure 42. Prime family 197
2979
965 1337 3077 3401
199p
Figure 43. Prime family 199
4941
338 2189 2561 3281 3629
211p
Figure 44. Prime family 211
562
284
220
Figure 45. Aliquot cycle 220
2321 4577
223p
Figure 46. Prime family 223 26 RICHARD J. MATHAR
4627
669
227p
Figure 47. Prime family 227
1115 3587
229p
Figure 48. Prime family 229
4581
2049
687 1135
233p
Figure 49. Prime family 233
762
774
942
954
1152
2163
2497 1165
239p
Figure 50. Prime family 239 ALIQUOT SEQUENCE MERGERS 27
2905 3369 4881
399 1127 1631 2519 2951 3791
241p
Figure 51. Prime family 241 28 RICHARD J. MATHAR
2629 3961
251p
Figure 52. Prime family 251
4292
3688
3242 4910 4730
1624 3946 3932 4774
1976 2956 4442
2224 4226
2116
1755 2499 3723 4803
1605 2949
987 1131 3787 4697 4232
549 1255 4063
257p
Figure 53. Prime family 257
1285 2761
263p
Figure 54. Prime family 263 ALIQUOT SEQUENCE MERGERS 29
2093 4109 2888
595 1315 2827 4267
269p
Figure 55. Prime family 269
3483 4113
1841 3341 4769
271p
Figure 56. Prime family 271
276 306
396
696
1104
3430 1872
3770
3790
2864 3050
2716
2772
5964
Figure 57. Sequence of 276 with unknown termination. 30 RICHARD J. MATHAR
3063
1025 1355 1883 3419 4883
277p
Figure 58. Prime family 277
3651 2955 4779
1221 1797 2481
603 831 1183 2959 4471
281p
Figure 59. Prime family 281
4143
1385 2981 3497 4997
283p
Figure 60. Prime family 283
3545
715 3091
293p
Figure 61. Prime family 293
3809 4913
307p
Figure 62. Prime family 307 ALIQUOT SEQUENCE MERGERS 31
2043
921 4981
311p
Figure 63. Prime family 311
3513 4593
1175 1535
313p
Figure 64. Prime family 313
939 1555
317p
Figure 65. Prime family 317
4121
331p
Figure 66. Prime family 331
1655 1859 2057
337p
Figure 67. Prime family 337
805
347p
Figure 68. Prime family 347 32 RICHARD J. MATHAR
324
523p
Figure 69. Prime family 523 ALIQUOT SEQUENCE MERGERS 33
180 210 354
366
378
822 582
834 594
612 846
930 1362 1026
1374
2262 2766 1386
2778 2358
2790
4698
6192
Figure 70. Prime family 601 34 RICHARD J. MATHAR
120
240
504
1272 1056
1968
3240
7650
Figure 71. Prime family 12161 ALIQUOT SEQUENCE MERGERS 35
318
498 330
510 534
786 546
750 798
1122
1830 1974 1470
2634
2646
4194
4932
7626
Figure 72. Prime family 321329 36 RICHARD J. MATHAR
3. Unitary Sigma Illustrations Mapping n to the sum of the unitary divisors of n [2,3,4] as tabulated in A034448 [9] defines another set of merger families, illustrated in Figures 73–77. ALIQUOT SEQUENCE MERGERS 37 3 6 4 2 5 12 20 11 19 29 30 16 17 18 71 8 9 7 10 55 51 72 46 89 90 39 56 58 179 68 67 47 153 35 48 180 31 145 32 33 261 21 118 300 151 245 152 475 236 111 520 412 403 697 227 448 273 689 185 664 756 316 1120 187 251 228 502 184 221 400 442 159 205 252 109 409 410 142 166 79 101 216 110 265 63 80 324 102 214 99 57 130 95 120 92 100 169 170 59 87 119 45 60 115 43 44 144 76 94 38 37 66 52 70
Figure 73. Unitary sigma family 2 38 RICHARD J. MATHAR 13 14 23 24 36 50 22 78 15 49 167 168 143 253 135 235 123 213 288 183 248 330 864 218 165 154 138 137
Figure 74. Unitary sigma family 13 ALIQUOT SEQUENCE MERGERS 39 26 25 42 41 77 96 69 132 131 62 61 86 239 209 207 203 157 188 240 139 177 117 107 158 88 108 140 85 125 126 368 81 82 408 335 103 104 114 647 303 75 367 113 583 73 74 568 648 738 1260 373 535 439 374 387 440 327 318 317
Figure 75. Unitary sigma family 25 40 RICHARD J. MATHAR 28 40 27 53 54 34 83 84 64 65 133 160 198 197 124 359 323 319 295 360 283 284 493 267 173 174 477 172 220 472 190 148 445 540 840 204 134 428 127 128 129 176 358 93 289 290 268 340 259 304 306 202 271 272 201
Figure 76. Unitary sigma family 27 ALIQUOT SEQUENCE MERGERS 41 91 136 112 246 162 106 503 459 451 415 504 375 269 334 719 232 286 667 212 211 270 649 178 639 311 623 312 275 537 149 206 495 150 116 478 98 97 431 631 720 1020 632 391 471 460 376 435 371 476 355 418 432 321 229 406 255 353 263 354 230 274 414 223 215 264 407 189 224 238 208 339 456 175 280 247 302 146 222 156
Figure 77. Unitary sigma family 91 42 RICHARD J. MATHAR
References 1. Manuel Benito and Juan L. Varona, Advances in aliquot cycles, Math. Comp. 68 (1999), no. 225, 389–393. MR 1489967 (99c:11162) 2. Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Zeitschr. 74 (1960), 66–80. MR 0112861 (22#3707) 3. , The number of unitary divisors of an integer, Am. Math. Monthly 67 (1960), no. 9, 879–880. MR 0122790 (23#A124) 4. Graeme L. Cohen, On an integer’s infinitary divisors, Math. Comp. 54 (1990), no. 189, 395– 411. MR 0993927 (90e:11011) 5. P. Erd˝os, On asymptotic properties of aliquot sequences, Math. Comp. 30 (1976), no. 135, 641–645. MR 0404115 (53#7919) 6. Richard K. Guy, Unsolved problems in number theory, 3 ed., Springer, Heidelberg, 2004. MR 2076335 (2005h:11003) 7. Richard K. Guy and J. L. Selfridge, What drives an aliquot sequence?, Math. Comp. 29 (1975), no. 129, 101–107. MR 0384669 (52#5542) 8. David Moews and Paul C. Moews, A search for aliquot cycles and amicable pairs, Math. Comp. 61 (1993), no. 204, 935–938. MR 1185249 (94a:11198) 9. Neil J. A. Sloane, The On-Line Encyclopedia Of Integer Sequences, Notices Am. Math. Soc. 50 (2003), no. 8, 912–915, http://oeis.org/. MR 1992789 (2004f:11151) URL: http://www.mpia.de/~mathar
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