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betrothed pairs
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 
2526474 2 
34944502 3 
44774449430 4 
5516352831390 5 
61022422392280 6 
74226648684759290 7 
833125211351461928295 8 
927831189701161191268 9 
101832115929233314132298 10 
11273766560639074616770101

A pictorial representation of the table above
motab
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.