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 |
2 | 7 | 9951 | 2 |
3 | 6 | 9952 | 0 | 3 |
4 | 6 | 9951 | 1 | 0 | 4 |
5 | 3 | 5386 | 1814 | 1857 | 898 | 5 |
6 | 1 | 9946 | 0 | 5 | 6 | 0 | 6 |
7 | 7 | 2043 | 1884 | 1325 | 2038 | 1314 | 1347 | 7 |
8 | 5 | 5080 | 0 | 0 | 1 | 4871 | 1 | 0 | 8 |
9 | 4 | 3442 | 0 | 1 | 3267 | 0 | 1 | 3243 | 0 | 9 |
10 | 0 | 5382 | 0 | 1854 | 0 | 3 | 4 | 1814 | 3 | 898 | 10 |
11 | 1 | 1073 | 847 | 1068 | 1135 | 1123 | 960 | 882 | 931 | 1070 | 868 |
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.