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hyperperfect numbers
A number n such that n = 1 + k ⋅ (σ(n) - n - 1), for some k. more

The first 600 hyperperfect numbers :
6, 21, 28, 301, 325, 496, 697, 1333, 1909, 2041, 2133, 3901, 8128, 10693, 16513, 19521, 24601, 26977, 51301, 96361, 130153, 159841, 163201, 176661, 214273, 250321, 275833, 296341, 306181, 389593, 486877, 495529, 542413, 808861, 1005421, 1005649, 1055833, 1063141, 1232053, 1284121, 1403221, 1433701, 1570153, 1618597, 1787917, 1950625, 2287933, 2462881, 2469601, 2488201, 2666437, 2708413, 2768581, 2856481, 2884501, 2924101, 3328921, 3420301, 4013833, 4304341, 4312681, 4658449, 5199013, 6392257, 7152001, 7478041, 8766061, 8883841, 9398593, 9427657, 9699181, 10445221, 12064333, 12160249, 12283693, 13646989, 14444041, 14489437, 15042553, 15407173, 16260901, 16641241, 16904101, 18116737, 19106449, 20103001, 20151517, 20419093, 21396313, 23548753, 25773901, 26249473, 26585233, 28005301, 28449901, 28600321, 29751229, 29757601, 30374941, 31696501, 31837801, 32545741, 33550336, 34251601, 35315221, 35492533, 35494981, 36171853, 36640993, 37021141, 38159521, 38749153, 39116221, 39147301, 40374013, 41354041, 41881201, 42387097, 46024681, 46596457, 47014801, 47842237, 48373597, 50809261, 52606861, 53120149, 54296677, 55262737, 56979661, 59675773, 60110701, 61442077, 61599553, 62722153, 62804941, 63519481, 64220161, 65618101, 77939221, 79001833, 79089541, 81329701, 81705601, 88347781, 91736353, 94126657, 94472041, 95235601, 95295817, 97438381, 97585249, 101402761, 105070201, 106875541, 108240841, 112803841, 112891993, 113513761, 115788961, 116128021, 118535929, 119313541, 124035913, 125150281, 125438101, 127180033, 127802533, 128990941, 129127041, 132123973, 133815229, 134281141, 134927281, 134959501, 138167929, 138786229, 140520901, 147816121, 150366841, 151547761, 153346261, 155367733, 156796501, 157026301, 157767481, 158048281, 158813737, 160123261, 162157789, 163331269, 168159889, 170023489, 176344741, 179920057, 181132801, 181728817, 185560633, 186898501, 190986001, 191535121, 191847181, 192317941, 194283181, 196581361, 199839457, 204349393, 205100017, 205253533, 214696441, 217033693, 219575101, 221814409, 228382837, 228695497, 229971241, 234784201, 243000001, 243328381, 244930669, 247005973, 247941649, 255286081, 258051649, 264521701, 265434901, 267639481, 273239821, 275797861, 281517121, 289054369, 291235093, 291461701, 302343133, 312210169, 313255213, 318561193, 333080509, 346548133, 350992501, 353944177, 357858001, 359269021, 362320489, 375014317, 383550289, 383712781, 391854937, 395421517, 398383441, 401039341, 404458477, 407721793, 414857713, 417439801, 419745517, 420562573, 421569877, 430920001, 432373033, 434307901, 438281281, 440179273, 441116281, 445057057, 447422401, 448644961, 454166017, 454503817, 470912293, 476610241, 483805033, 495196453, 504662701, 509031157, 511017757, 512459221, 517093141, 528316933, 528793501, 529563217, 530710561, 537399589, 540811477, 542935873, 551522161, 551800621, 559305469, 569450701, 571145173, 573685201, 578710501, 579440497, 584116261, 585079177, 609745501, 611417641, 612302101, 615143437, 623276617, 639596281, 646669549, 652455757, 659185621, 661770013, 662313601, 665453881, 668541001, 682591453, 688773781, 690653329, 705462181, 724525861, 737640901, 744908701, 749071753, 749797501, 749931337, 750787501, 753152557, 761997661, 762484417, 775307917, 775316593, 796622401, 801416221, 808275241, 809979901, 819268081, 822000961, 824815681, 828884629, 836729233, 837769213, 848374801, 852687061, 853613713, 867716581, 871847101, 877148941, 893748277, 894188761, 914054329, 922604701, 924642337, 925572421, 939137257, 940209541, 941755189, 969890533, 972261181, 974380921, 1007427661, 1014428773, 1027025941, 1056903469, 1060913701, 1063290841, 1066544641, 1076086393, 1104828061, 1117454893, 1118457481, 1131768301, 1133270353, 1160283217, 1167773821, 1170219493, 1182367117, 1192769533, 1194832213, 1196470021, 1205699401, 1208362789, 1215267817, 1218260233, 1220640625, 1238298121, 1246336981, 1266324793, 1304557801, 1305228121, 1331220601, 1339967137, 1356240757, 1367543809, 1368482593, 1368950977, 1369932013, 1378737469, 1383381661, 1388292601, 1392422041, 1398088513, 1408012273, 1411495321, 1414767433, 1419646981, 1420285033, 1431684853, 1445322817, 1451334757, 1455503893, 1462960909, 1463397769, 1466992081, 1467111661, 1477109869, 1477518073, 1483186741, 1490040301, 1528625113, 1533533893, 1537161889, 1538669401, 1539180673, 1543705489, 1564317613, 1566930901, 1589656729, 1603199197, 1606119301, 1611546241, 1620278281, 1632665701, 1646586817, 1652188201, 1659368101, 1661570413, 1662667093, 1667411377, 1674741469, 1675725397, 1704396697, 1711192501, 1715479033, 1730882401, 1732504093, 1751757253, 1792654141, 1822137277, 1846006501, 1857884857, 1863895261, 1864137277, 1877476801, 1882359229, 1899051937, 1909488457, 1930480897, 1939497541, 1964378569, 1969584301, 1970278213, 1970692441, 1977225901, 1991372137, 2000111881, 2002872577, 2008230001, 2035095973, 2054553001, 2056638481, 2061046237, 2068825201, 2069521837, 2072296201, 2073239593, 2073781441, 2082096901, 2098966957, 2123442361, 2144851153, 2150435269, 2157934237, 2165135641, 2177876101, 2193640177, 2204526001, 2274834193, 2288948341, 2324355601, 2326434961, 2340579721, 2341431361, 2349167341, 2363709037, 2373946321, 2398893421, 2410440481, 2434089997, 2436228181, 2441394541, 2446606501, 2449628917, 2469439417, 2470988053, 2498361721, 2507026741, 2513906761, 2514440713, 2528766181, 2555170021, 2579163421, 2613283681, 2615412601, 2618199301, 2619914701, 2631883561, 2634645001, 2653061533, 2657468641, 2662218553, 2663019001, 2666316997, 2677161433, 2681428777, 2685007417, 2687834557, 2725991209, 2728562689, 2731565701, 2765559157, 2770452001, 2771077501, 2797540201, 2840661361, 2842794469, 2879450461, 2880630097, 2889150193, 2902224361, 2904313057, 2923704001, 2929971601, 2934414661, 2938022701, 2945287129, 2951097601, 3002075521, 3036787153, 3037474597, 3064960861, 3065836801, 3067705117, 3069636481, 3073798633, 3074660929, 3082241917, 3102982261, 3120085009, 3121581217, 3147679309, 3159549901, 3188608561, 3200766253, 3216287629, 3239559493, 3265302529, 3288287557, 3289452217, 3313311061, 3349119313, 3352733641, 3396199717, 3399557149, 3400141861, 3412327381, 3413722681, 3426618541, 3470437921, 3483941761, 3489409501, 3536062741, 3561945949, 3571115101, 3581333137, 3609413893, 3632227957, 3635931973, 3660101413, 3662246773, 3663498721, 3681455017, 3689586673, 3692421217, 3693025057, 3714274297, 3719738881, 3769878613, 3777981481, 3801120481, 3806207761, 3839454481, 3845297341, 3849183157, 3849202501, 3857191813, 3889272301, 3922508509, 3958311793.

Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 9958 values, from 6 to 3999666429301).

n\r 0  1 
279951 2 
3699520 3 
46995110 4 
53538618141857898 5 
6199460560 6 
77204318841325203813141347 7 
855080001487110 8 
9434420132670132430 9 
10053820185403418143898 10 
11110738471068113511239608829311070868

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.