Quasi-aliquot Sequences?

sweety439 · 2019-07-14, 09:04

In mathematics, a "quasiperfect number" is a natural number n for which n is the sum of its non-trivial divisors (i.e., its divisors excluding 1 and n), similarly, we define "quasi-aliquot sequence", a quasi-aliquot sequence is a sequence of positive integers in which each term is the sum of the non-trivial divisors (i.e., its divisors excluding 1 and n) of the previous term. e.g. the quasi-aliquot sequence of 36 is 36, 54, 65, 18, 20, 21, 10, 7, 0. Does quasi-aliquot sequence always end with either 0 or quasi-amicable pair (betrothed pair, such as 48 and 75)?

garambois · 2019-07-14, 10:12

I know at least one more quasi 8-cycles :

0 1270824975
1 1467511664
2 1530808335
3 1579407344
4 1638031815
5 1727239544
6 1512587175
7 1215571544
8 1270824975
9 1467511664
10 1530808335
11 1579407344
...
...

I had done many tests a few years ago with different iterative processes :

n --> s(n) + b

s(n)=sigma(n)-n
If b=0 : aliquot sequences
If b=-1 : quasi-aliquot sequences

For example, if b=38, it exists a 298-cycle !!!

You can see on this page, but sorry, in french (s(n) is called sigma'(n) on this page) : http://www.aliquotes.com/autres_proc...iteratifs.html

I had even tried to extend the sigma function to something other than integers, such as polynomials, or Gauss integers. But the problem is that there is always something of the conventional order in these last cases, see here : http://www.aliquotes.com/etendre_sigma.html

sweety439 · 2020-06-21, 21:45

Conjecture: there are no quasi n-cycles if n is odd, specially, there are no quasi-perfect numbers.

sweety439 · 2022-08-04, 09:17

I have found that: the sequence a(n+1) = s(a(n))+k, start with any prime number:

Code:

1: 2,
2: 3,
3: 4, 6, 9, 7,
4: 5,
5: 6, 11,
6: 7,
7: 8, 14, 17,
8: 9, 12, 24, 44, 48, 84, 148, 126, 194, 108, 180, 374, 282, 302, 162, 209, 39, 25, 14, 18, 29,
9: 10, 17,
10: 11,
11: 12, 27, 24, 47,
12: 13,
13: 14, 23,
14: 15, 23,
15: 16, 30, 57, 38, 37,
16: 17,
17: 18, 38, 39, 34, 37,
18: 19,
19: 20, 41,
20: 21, 31,
21: 22, 35, 34, 41,
22: 23,
23: 24, 59,
24: (infinite, aperiodic)
25: 26, 41,
26: 27, 39, 43,
27: 28, 55, 44, 67,
28: 29,
29: 30, 71,
30: 31,
31: 32, 62, 65, 50, 74, 71,
32: 33, 47,
33: 34, 53,
34: 35, 47,
35: 36, 90, 179,
36: 37,
37: 38, 59,
38: 39, {55}
39: 40, 89,
40: 41,
41: 42, 95, {66, 119}
42: 43,
43: 44, 83,
44: 45, 77, 63, 85, 67,
45: 46, 71,
46: 47,
47: 48, {123, 92}
48: 49, 56, 112, 184, 224, 328, 350, 442, 362, 232, 266, 262, 182, 202, 152, 196, 251,
49: 50, 92, 125, 80, 155, 86, 95, 74, 89,
50: 51, 71,
51: 52, 97,
52: 53,
53: 54, 119, {78, 143}
54: 55, 71,
55: 56, 119, 80, 161, 86, 101,
56: 57, 79,
57: 58, 89,
58: 59,
59: 60, 167,
60: 61,
61: 62, 95, 86, 107,
62: 63, 103,
63: 64, 126, 249, 150, 285, 258, 333, 224, 343, 120, 303, 168, 375, 312, 591, 264, 519, 240, 567, 464, 529, 87, 96, 219, 140, 259, 108, 235, 116, 157,
64: 65, 83,
65: 66, 143, 90, 209, {96, 221}
66: 67,
67: 68, 125, 98, 140, 263,
68: 69, 95, 93, 103,
69: 70, 143, {94, 119}
70: 71,
71: 72, 194, 171, 160, 289, 89,
72: 73,
73: 74, 113,
74: 75, 123, 119, 99, 131,
75: 76, 139,
76: 77, 95, 101,
77: 78, 167,
78: 79,
79: 80, {185, 122, 143, 104}
80: (infinite, aperiodic)
81: 82, 125, 112, 217, 120, 321, 192, 397,
82: 83,
83: 84, 223,
84: 85, 107,
85: 86, 131,
86: 87, 119, 111, 127,
87: 88, 179,
88: 89,
89: 90, 233,
90: 91, 111, 131,
91: 92, 167,
92: 93, 127,
93: 94, 143, 118, 155, 130, 215, 142, 167,
94: 95, {119}
95: 96, 251,
96: 97,
97: 98, 170, 251,
98: 99, 155, {135, 203}
99: 100, 216, 483, 384, 735, 732, 1103,
100: 101,
101: 102, 215, 150, 323, 138, 251,
102: 103,
103: 104, 209, 134, 173,
104: 105, 191,
105: 106, 161, 136, 239,
106: 107,
107: 108, 279, 244, 297, 290, 357, 326, 273, 282, 401,
108: 109,
109: 110, 215, 158, 191,
110: 111, 151,
111: 112, 247, 144, 370, 425, 244, 301, 162, 312, 639, 408, 783, 528, 1071, 912, 1679, 208, 337,
112: 113,
113: 114, 239,
114: 115, 143, 139,
115: 116, 209, 146, 191,
116: 117, 181,
117: 118, 179,
118: 119, {143}
119: 120, 359,
120: 121, 132, 324, 643,
121: 122, 185, 164, 251,
122: 123, 167,
123: 124, 223,
124: 125, {155, 161}
125: 126, 311,
126: 127,
127: 128, 254, 257,
128: 129, 175, 201, 199,
129: 130, 251,
130: 131,
131: 132, 335, 204, 431,
132: 133, 159, 189, 263,
133: 134, 203, 170, {287, 182}
134: 135, 239,
135: 136, 269,
136: 137,
137: 138, 287, 186, 335, 210, 503,
138: 139,
139: 140, {335, 212, 305, 206, 245, 236, 323, 176}
140: 141, 191,
141: 142, 215, 190, 311,
142: 143, 167,
143: 144, 402, 557,
144: 145, 179,
145: 146, 221, 176, 341, 188, 293,
146: 147, 227,
147: 148, 265, 206, 253, 182, 301, 198, 417, 290, 397,
148: 149,
149: 150, 371, 210, 515, 258, 419,
150: 151,
151: 152, {299, 188}
152: 153, 233,
153: 154, 287, 202, 257,
154: 155, 191,
155: 156, 391, 196, 358, 337,
156: 157,
157: 158, 239,
158: 159, 215, 207, 263,
159: 160, 377, 202, 263,
160: 161, 191,
161: 162, 362, 345, 392, 624, 1273, 248, 393, 296, 435, 446, 387, 346, 337,
162: 163,
163: 164, 293,
164: 165, 287, 213, 239,
165: 166, 251,
166: 167,
167: 168, 479,
168: 169, 182, 322, 422, 382, 362, 352, 572, 772, {754, 674, 508, 556, 592}
169: 170, 323, 206, 275, 266, 383,
170: 171, 259, 215, 219, 247, 203, 207, 275, 267, 263,
171: 172, 307,
172: 173,
173: 174, 359,
174: 175, 247, 207, 279, 311,
175: 176, 371, 236, 359,
176: 177, 239,
177: 178, 269,
178: 179,
179: 180, 545, 294, 569,
180: 181,
181: 182, 335, 254, 311,
182: 183, 247, 215, 231, 335, 255, 359,
183: 184, 359,
184: 185, 227,
185: 186, 383,
186: 187, 215, 235, 239,
187: 188, 335, 260, 515, 296, 461,
188: 189, 319, 229,
189: 190, 359,
190: 191,
191: 192, 507, 416, 657, 496, 687, 424, 577,
192: 193,
193: 194, 293,
194: 195, 335, 267, 287, 243, 315, 503,
195: 196, 398, 397,
196: 197,
197: 198, 467,
198: 199,
199: 200, 464, 665, 494, 545, 314, 359,
200: 201, 271,
201: 202, 305, 268, 409,
202: 203, 239,
203: 204, 503,
204: 205, 251,
205: 206, 311,
206: 207, 311,
207: 208, 433,
208: 209, 239,
209: 210, 575, 378, 791, 330, 743,
210: 211,
211: 212, 377, {254, 341}
212: 213, 287, 261, 341, 255, 389,
213: 214, 323, 250, 431,
214: 215, 263,
215: 216, 599,
216: 217, 255, 393, 351, 425, 349,
217: 218, 329, 272, 503,
218: 219, 295, 283,
219: 220, 503,
220: 221, 251,
221: 222, 455, 438, 671, 294, 611, 282, 515, 330, 755, 378, 803, 306, 617,
222: 223,
223: 224, 503,
224: (infinite, aperiodic)
225: 226, 341, 268, 433,
226: 227,
227: 228, 559, 284, 447, 380, 687, 460, 775, {444, 847}
228: 229,
229: 230, 431,
230: 231, 383,
231: 232, 449,
232: 233,
233: 234, 545, 348, 725, 438, 683,
234: 235, 287, 283,
235: 236, 419,
236: 237, 319, 277,
237: 238, 431,
238: 239,
239: 240, 743,
240: 241,
241: 242, 398, 443,
242: 243, 363, 411, 383,
243: 244, 433,
244: 245, 341, 287, 293,
245: 246, 503,
246: 247, 279, 383,
247: 248, 479,
248: 249, 335, 321, 359,
249: 250, 467,
250: 251,
251: 252, 727,
252: 253, 287, 301, 303, 357, 471, 413, 319, 293,
253: 254, 383,
254: 255, 431,
255: 256, 510, 1041, 606, 873, 656, 901, 326, 421,
256: 257,
257: 258, 527, 306, 653,
258: 259, 303, 363, 427, 327, 371, 319, 299, {295, 323}
259: 260, 587,
260: 261, 389,
261: 262, 395, 346, 437, 304, 577,
262: 263,
263: 264, 719,
264: 265, 323, 301, 315, 573, 459, 525, 731, 325, 373,
265: 266, 479,
266: 267, 359,
267: 268, 475, 412, 583, 332, 523,
268: 269,
269: 270, 719,
270: 271,
271: 272, 557,
272: 273, 447, 425, 405, 593,
273: 274, 413, {340, 689}
274: 275, 371, 335, 347,
275: 276, 671, {348, 767}
276: 277,
277: 278, 419,
278: 279, 415, 367,
279: 280, 719,
280: 281,
281: 282, 575, 450, 1040, 1845, 1712, 1917, 1244, 1221, 884, 1161, 880, 1633, 376, 625, 437, 324, 804, 1381,
282: 283,
283: 284, 503,
284: 285, 479,
285: 286, 503,
286: 287, 335, 359,
287: 288, 818, 699, 524, 687, 520, 1027, 380, 747, 632, 855, 992, 1311, 896, 1431, 1016, 1191, 688, 963, 728, 1239, 968, 1314, 1859, 624, 1399,
288: 289, 306, 684, 1424, 1654, 1118, 1018, 800, 1441, 431,
289: 290, 539, 434, 623, 386, 485, 392, 752, 1025, 566, 575, 458, 521,
290: 291, 391, 331,
291: 292, 517, 350, 685, 434, 625, 447, 444, 911,
292: 293,
293: 294, 683,
294: 295, 359,
295: 296, 569,
296: 297, 479,
297: 298, 449,
298: 299, 335, 371, 359,
299: 300, 867, 660, 1655, 636, 1175, 612, 1325, 648, 1466, 1035, 1136, 1395, 1400, 2619, 1600, 2636, 2283, 1064, 1635, 1304, {1455, 1196}
300: 301, 351, 509,
301: 302, 455, 518, {695, 446, 527, 350}
302: 303, 407, 351, 511, 383,
303: 304, 619,
304: 305, 371, 365, 383,
305: 306, 701,
306: 307,
307: 308, 671, {380, 767}
308: 309, 415, 397,
309: 310, 575, 478, 551, 358, 491,
310: 311,
311: 312, 839,
312: 313,
313: 314, 473, 368, 689, 380, 773,
314: 315, 623, 411, 455, 531, 563,
315: 316, 559, 372, 839,
316: 317,
317: 318, 647,
318: 319, 359,
319: 320, 761,
320: 321, 431,
321: 322, 575, 490, 857,
322: 323, 359,
323: 324, 846, 1349, 414, 845, 576, 1398, 1733,
324: 325, 433,
325: 326, 491,
326: 327, 439,
327: 328, 629, 382, 521,
328: 329, 383,
329: 330, 863,
330: 331,
331: 332, 587,
332: 333, 493, 379,
333: 334, 503,
334: 335, 407, 383,
335: 336, 991,
336: 337,
337: 338, 548, 755, 494, 683,
338: 339, 455, 555, 695, 483, {623, 435}
339: 340, 755, 496, 835, 512, 850, 1163,
340: 341, 383,
341: 342, 779, {402, 755, 498, 851}
342: 343, 399, 583, 407, 391, 383,
343: 344, 659,
344: 345, 575, 513, 631,
345: 346, 521,
346: 347,
347: 348, 839,
348: 349,
349: 350, 743,
350: 351, 559, 407, {399, 591, 551}
351: 352, 755, 508, 739,
352: 353,
353: 354, 719,
354: 355, 431,
355: 356, 629, 410, 701,
356: 357, 575, 525, 823,
357: 358, 539, 502, 611, 418, 659,
358: 359,
359: 360, 1169, 534, 905, 546, 1157, 462, 1049,
360: 361, 380, 820, 1304, 1516, 1504, 1880, 2800, 5248, 5822, 3610, 3608, 4312, 6308, 5812, 4726, 3194, 1960, 3530, 3202, 1964, 1840, 2984, 2986, 1856, 2314, 1826, 1558, 1322, 1024, 1383, 825, 1023, 873, 761,
361: 362, 545, 476, 893, {428, 689}
362: 363, 531, 611, 423, 563,
363: 364, 783, 780, 1935, 1860, 3879, 2100, 5207, 532, 951, 684, 1499,
364: 365, 443,
365: 366, 743,
366: 367,
367: 368, 743,
368: 369, 545, 483, 653,
369: 370, 683,
370: 371, 431,
371: 372, 895, 556, 795, 872, 1149, 758, 753, 626, 687, 604, 831, 652, 867, 732, 1375, 868, 1295, 900, 2292, 3455, 1068, 1823,
372: 373,
373: 374, 647,
374: 375, 623, 471, 535, 487,
375: 376, 719,
376: 377, 419,
377: 378, 959, 522, 1025, 654, 1043, 534, 923, 462, 1067, 486, 983,
378: 379,
379: 380, 839,
380: 381, 511, 461,
381: 382, 575, 550, 947,
382: 383,
383: 384, 1019,
384: 385, 575, 553, 471, 545, 499,
385: 386, 581, 476, 917, 524, 785, 548, 803, 470, 779, {446, 611}
386: 387, 571,
387: 388, 685, 530, 829,
388: 389,
389: 390, 1007, 462, 1079, 486, 995, 594, {1235, 834}
390: 391, 431,
391: 392, 854, 1025, 668, 899, 452, 737, 470, 785, 554, 671, 464, 857,
392: 393, 527, 441, 692, 918, 1634, 1398, 1802, 1506, 1910, 1938, 2774, 2058, 3134, 1962, 2720, 4476, 6388, 5190, 7730, 6594, 8966, 4878, 6122, 3456, 7136, 7368, 11504, 11208, 17264, 19584, 40478, 22386, 34454, 28146, 28550, 25038, 34322, 17949, 7275, 5269, 883,
393: 394, 593,
394: 395, 479,
395: 396, 1091,
396: 397,
397: 398, 599,
398: 399, 639, 695, 543, 583, 463,
399: 400, 960, 2487, 1232, 2143,
400: 401,

Conjecture: If k+1 is not odd square number, then the sequence must terminate at either a prime or a cycle

2019-07-14, 09:04	#1
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2⁵·5·23 Posts	Quasi-aliquot Sequences? In mathematics, a "quasiperfect number" is a natural number n for which n is the sum of its non-trivial divisors (i.e., its divisors excluding 1 and n), similarly, we define "quasi-aliquot sequence", a quasi-aliquot sequence is a sequence of positive integers in which each term is the sum of the non-trivial divisors (i.e., its divisors excluding 1 and n) of the previous term. e.g. the quasi-aliquot sequence of 36 is 36, 54, 65, 18, 20, 21, 10, 7, 0. Does quasi-aliquot sequence always end with either 0 or quasi-amicable pair (betrothed pair, such as 48 and 75)? Last fiddled with by sweety439 on 2019-07-14 at 09:13

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2019-07-14, 10:12	#2
garambois "Garambois Jean-Luc" Oct 2011 France 2×17×31 Posts	I know at least one more quasi 8-cycles : 0 1270824975 1 1467511664 2 1530808335 3 1579407344 4 1638031815 5 1727239544 6 1512587175 7 1215571544 8 1270824975 9 1467511664 10 1530808335 11 1579407344 ... ... I had done many tests a few years ago with different iterative processes : n --> s(n) + b s(n)=sigma(n)-n If b=0 : aliquot sequences If b=-1 : quasi-aliquot sequences For example, if b=38, it exists a 298-cycle !!! You can see on this page, but sorry, in french (s(n) is called sigma'(n) on this page) : http://www.aliquotes.com/autres_proc...iteratifs.html I had even tried to extend the sigma function to something other than integers, such as polynomials, or Gauss integers. But the problem is that there is always something of the conventional order in these last cases, see here : http://www.aliquotes.com/etendre_sigma.html

2020-06-21, 21:45	#3
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2⁵×5×23 Posts	Conjecture: there are no quasi n-cycles if n is odd, specially, there are no quasi-perfect numbers.