A member of a pair (n, m) such that there exist another number σ(n) - n - 1 = m and σ(m) - m - 1 = n. more
The first 600 betrothed numbers :
48,
75,
140,
195,
1050,
1575,
1648,
1925,
2024,
2295,
5775,
6128,
8892,
9504,
16587,
20735,
62744,
75495,
186615,
196664,
199760,
206504,
219975,
266000,
309135,
312620,
507759,
526575,
544784,
549219,
573560,
587460,
817479,
1000824,
1057595,
1081184,
1139144,
1140020,
1159095,
1173704,
1208504,
1233056,
1236536,
1279950,
1331967,
1341495,
1348935,
1459143,
1524831,
1763019,
1902215,
1921185,
2036420,
2102750,
2140215,
2142945,
2171240,
2198504,
2226014,
2312024,
2421704,
2576945,
2580864,
2681019,
2958500,
3010215,
3123735,
3220119,
3676491,
4012184,
4282215,
4311024,
5088650,
5416280,
5644415,
6081680,
6446325,
6618080,
7460004,
7509159,
7875450,
7890575,
8713880,
8829792,
9247095,
9345903,
10106504,
10925915,
12146750,
12251679,
12500865,
12900734,
13693959,
13922100,
14371104,
16247745,
16381925,
18845855,
22013334,
22559060,
23379224,
23939685,
26409320,
26502315,
26525415,
27735704,
27862695,
28206815,
28219664,
31213899,
31356314,
32014575,
33299000,
34093304,
37291625,
37324584,
40818855,
41137620,
41950359,
42125144,
43321095,
49217084,
52026920,
52389315,
52601360,
58354119,
61423340,
62252000,
64045904,
66086504,
66275384,
68337324,
69090615,
70112175,
72917000,
76011992,
77723360,
80870615,
84854315,
85141719,
87689415,
87802407,
88567059,
89446860,
93423519,
93993830,
94713300,
94970204,
96751395,
97389039,
97797104,
99735705,
100256480,
101366342,
101589320,
102019644,
105993657,
106004024,
106524264,
114332175,
114859884,
115780599,
116329136,
116481464,
116540304,
117106064,
117869940,
128707425,
128825415,
129081735,
130016744,
130304888,
133592175,
134064584,
139829624,
140687415,
141649235,
142287992,
145810719,
147041504,
150157215,
152340944,
153582624,
154043505,
154885920,
157470404,
157906430,
158948000,
160797560,
161733825,
161888544,
162156995,
164340224,
164799635,
165416048,
165507975,
166335255,
168102044,
168669879,
170262560,
171553635,
172622505,
173431335,
174080799,
175742294,
178415048,
183651663,
184191111,
189208575,
191800455,
193773447,
195318500,
197570264,
197845235,
198727544,
198978584,
199563195,
201006414,
204064175,
205556894,
207351704,
208548624,
210391694,
210533904,
214559631,
215676320,
226351671,
226964240,
227323250,
227437280,
230092730,
231472535,
233348655,
237110120,
237316959,
237435495,
240536075,
242854884,
248611815,
252734468,
260310375,
261335025,
261412235,
262754384,
265664175,
268315515,
269436699,
271842885,
274313564,
281885919,
285695190,
290057624,
293057960,
296092784,
300215895,
308040104,
308322399,
311172785,
313001199,
313472880,
316305549,
322435295,
335622279,
338626574,
341660220,
344163140,
348865935,
350145775,
350646435,
353587304,
354818060,
367762395,
374770655,
376157079,
379137590,
380747960,
390655395,
396897759,
400950459,
401836344,
407482460,
417152295,
419396439,
420556695,
421077215,
428870144,
436621604,
442147544,
443390409,
447538784,
449036315,
460824848,
478571775,
479527244,
486142620,
499295384,
506589020,
508826439,
516860015,
530004915,
548544744,
559939625,
579753384,
592455015,
597781820,
599291000,
627398240,
638390511,
645632204,
653318420,
654929715,
659384739,
662138295,
665749784,
676362435,
713342504,
721644104,
724235784,
726558860,
728192415,
734590850,
740532716,
754727336,
783839264,
787047488,
794572779,
794589635,
816390819,
817931828,
820888964,
844622295,
851088615,
856444875,
882424179,
888514263,
902555024,
921943035,
926979200,
932552445,
933562055,
949977644,
951044031,
953056175,
959131635,
980389395,
987558704,
987827555,
991087064,
1010000396,
1023736328,
1033002620,
1033763744,
1035813500,
1051937271,
1059627800,
1098115136,
1101593295,
1106039775,
1123418384,
1131760935,
1141074704,
1145890340,
1151128055,
1154706804,
1155794079,
1174809285,
1194885075,
1199626119,
1212941114,
1217353016,
1224278615,
1226825000,
1252992015,
1261742180,
1265855255,
1283921184,
1298344383,
1307247711,
1312252784,
1343056365,
1351707560,
1366942575,
1370080899,
1378574624,
1406830964,
1450207250,
1467581780,
1473700515,
1488607820,
1489939143,
1506753884,
1516137615,
1517220584,
1526208579,
1526672420,
1531953080,
1532741210,
1532917659,
1547906100,
1559119688,
1560890925,
1648746735,
1654350615,
1679061285,
1682066420,
1699088859,
1704525074,
1727172314,
1741044500,
1742984919,
1764267764,
1778869064,
1782222399,
1783523511,
1795408875,
1816402964,
1840746500,
1858726155,
1862919435,
1883876000,
1903187979,
1906051160,
1910285792,
1939995596,
1982390795,
1992599648,
1996209375,
2006332664,
2012227784,
2020337564,
2050561604,
2052672999,
2082593240,
2085372512,
2105199135,
2132355440,
2139198039,
2144382135,
2160543104,
2169354564,
2193824940,
2234106495,
2239676220,
2258708859,
2267171640,
2293334424,
2296563555,
2313337515,
2325991220,
2327248287,
2343404979,
2350626615,
2355309824,
2359000280,
2361854565,
2367009189,
2411826759,
2413813688,
2418204404,
2423454264,
2436393924,
2441496464,
2458574180,
2490395120,
2498946345,
2510904284,
2540165144,
2541717555,
2573840619,
2576707335,
2589127190,
2595146084,
2596144160,
2604140475,
2641212315,
2652982304,
2670130904,
2689435034,
2705286500,
2716915824,
2731700565,
2732097951,
2746338504,
2758505540,
2772801024,
2776954719,
2784075434,
2806455200,
2819077155,
2837602520,
2838165615,
2849128359,
2887857819,
2898322575,
2920453095,
2921793224,
2931650631,
2957731824,
3014994744,
3031422944,
3039432200,
3045615579,
3085130664,
3105475295,
3119339775,
3155451915,
3157070504,
3163223580,
3181556664,
3199177995,
3215269364,
3269502495,
3285728264,
3298072995,
3301952895,
3373298144,
3373643979,
3419884580,
3424407504,
3442577624,
3450876624,
3495040604,
3515685984,
3552265695,
3556889895,
3585589215,
3667278375,
3667561695,
3698827335,
3705257675,
3712611104,
3715037920,
3731353430,
3735363879,
3762043760,
3770681655,
3824655464,
3831852284,
3856140890,
3947516859,
3955132335,
3958280535,
3971454759,
3985532264,
3991840515,
3995575155,
4013193275,
4035324755,
4092444884,
4095184040,
4106225144,
4109899464,
4121667615,
4126705730,
4128822368,
4129427384,
4133786964,
4165376480,
4184346704,
4197999399,
4236950375,
4239746469,
4253975684,
4264348088,
4332612585,
4333216635,
4364776125,
4372365500,
4382760704,
4388870115,
4415176844,
4438526780,
4479663936,
4482524150,
4516899849,
4523120315,
4537964535,
4587544604,
4604412384,
4618069599,
4643378844,
4666112264,
4668784835,
4677638955,
4696878944,
4719190112,
4722238664,
4768579424,
4799782875,
4813637840,
4823747739,
4845714104,
4864759760,
4878780704,
4921491824,
4923510500,
4924118024,
4935148904,
4977874575,
5053883744.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 1000 values, from 48 to 20347923995).
n\r | 0 | 1 |
2 | 526 | 474 | 2 |
3 | 494 | 4 | 502 | 3 |
4 | 477 | 44 | 49 | 430 | 4 |
5 | 516 | 35 | 28 | 31 | 390 | 5 |
6 | 102 | 2 | 422 | 392 | 2 | 80 | 6 |
7 | 422 | 66 | 48 | 68 | 47 | 59 | 290 | 7 |
8 | 331 | 25 | 21 | 135 | 146 | 19 | 28 | 295 | 8 |
9 | 278 | 3 | 118 | 97 | 0 | 116 | 119 | 1 | 268 | 9 |
10 | 183 | 21 | 15 | 9 | 292 | 333 | 14 | 13 | 22 | 98 | 10 |
11 | 273 | 76 | 65 | 60 | 63 | 90 | 74 | 61 | 67 | 70 | 101 |
A pictorial representation of the table above
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.