A number such that the harmonic mean of its divisors is an integer. more
The first 600 harmonic divisor numbers :
1,
6,
28,
140,
270,
496,
672,
1638,
2970,
6200,
8128,
8190,
18600,
18620,
27846,
30240,
32760,
55860,
105664,
117800,
167400,
173600,
237510,
242060,
332640,
360360,
539400,
695520,
726180,
753480,
950976,
1089270,
1421280,
1539720,
2178540,
2229500,
2290260,
2457000,
2845800,
4358600,
4713984,
4754880,
5772200,
6051500,
8506400,
8872200,
11981970,
14303520,
15495480,
16166592,
17428320,
18154500,
23088800,
23569920,
23963940,
27027000,
29410290,
32997888,
33550336,
37035180,
44660070,
45532800,
46683000,
50401728,
52141320,
56511000,
69266400,
71253000,
75038600,
80832960,
81695250,
90409410,
108421632,
110583200,
115048440,
115462620,
137891520,
142990848,
144963000,
163390500,
164989440,
191711520,
221557248,
233103780,
255428096,
287425800,
300154400,
301953024,
318177800,
318729600,
326781000,
400851360,
407386980,
423184320,
428972544,
447828480,
459818240,
481572000,
499974930,
500860800,
513513000,
526480500,
540277920,
559903400,
623397600,
644271264,
675347400,
714954240,
758951424,
766284288,
819131040,
825120800,
886402440,
900463200,
995248800,
1047254400,
1162161000,
1199250360,
1265532840,
1307124000,
1352913408,
1379454720,
1381161600,
1509765120,
1558745370,
1630964808,
1632825792,
1727271000,
1862023680,
1867650048,
2008725600,
2140041600,
2144862720,
2369162250,
2481357060,
2701389600,
2705020500,
2716826112,
2738824704,
2763489960,
2777638500,
2839922400,
2876211000,
2945943000,
3134799360,
3209343200,
3221356320,
3288789504,
3328809120,
3349505250,
3506025600,
3594591000,
3702033720,
3740553180,
3831421440,
4143484800,
4146734592,
4720896180,
4738324500,
5058000640,
5133201408,
5275179000,
5297292000,
5510647296,
5579121240,
5943057120,
6720569856,
7279591410,
7330780800,
7515963000,
8104168800,
8154824040,
8243595360,
8410907232,
8436460032,
8589869056,
8628633000,
8659696500,
8696764800,
8698459616,
9866368512,
10200236032,
10575819520,
10597041000,
10597759200,
10952611488,
10983408128,
11076156000,
11296276992,
11480905800,
12941019000,
13067913600,
13073550336,
13398021000,
13581986600,
13584130560,
13660770240,
14182439040,
14254365440,
14378364000,
14541754500,
14980291200,
15174001920,
15192777600,
15358707000,
16003510272,
16569653760,
16919229600,
17624538624,
18999981000,
19017782784,
19209881600,
19744452000,
20015559200,
20387256120,
21537014400,
21611457280,
21943595520,
22633884000,
22933532160,
23450730240,
23855232960,
24362612820,
25559301600,
25666007040,
26113432800,
26242070400,
26454556800,
27122823000,
27689243400,
27726401736,
29715285600,
30063852000,
30600708096,
31638321000,
31727458560,
31766716800,
32950224384,
32956953120,
33040072800,
34174812672,
34482792960,
35032757760,
35793412200,
37906596000,
39970476000,
40053686400,
40520844000,
40752391680,
40805200800,
42054536160,
42763096320,
43783188480,
43861478400,
43952044500,
44184172032,
45578332800,
45923623200,
50497467930,
51001180160,
52748186400,
53227843200,
53621568000,
54572427000,
54648009000,
56481384960,
57575890944,
57629644800,
57648181500,
57897151488,
59388963480,
61434828000,
62487000576,
64834371840,
64914595200,
66433720320,
67622100480,
71271827200,
73924348400,
77120316000,
77924700000,
78340298400,
80422524000,
80533908000,
80551516500,
81417705600,
81488534400,
83410119000,
84418425000,
87825283840,
89526646440,
93419333280,
95088913920,
95300150400,
97941285120,
100383241728,
100522566144,
103262796000,
108061356200,
109111766400,
109585986048,
110886522600,
112202596352,
115987576320,
123014892000,
124406100000,
126090783000,
133410461184,
134369095680,
137438691328,
137770869600,
142275893760,
142985422944,
143173648800,
147112449120,
150115204512,
150759100800,
151955343540,
153003540480,
154567413000,
156473635500,
156798019840,
159248314400,
159381986400,
164297299320,
164751121920,
169696449000,
169956154368,
183694492800,
194743785600,
201532767744,
206166804480,
213815481600,
217494027520,
220524885504,
220920860160,
221908282624,
227783556000,
234605428736,
236489897160,
237191556096,
240423674400,
250230357000,
271309925250,
280541488500,
285266741760,
287879454720,
288662774400,
289048687200,
292337717760,
307001350656,
307030348800,
311203567584,
312402636000,
321300067176,
326196097920,
330097622400,
336607789056,
341519256000,
349002044160,
350280184800,
362526484320,
384342364800,
403031236608,
405280060416,
410240742912,
417624936960,
426778934400,
428440390560,
428555439000,
429520946400,
434508127200,
437409004032,
439655610240,
443622427776,
465036042240,
469420906500,
470717137800,
479411093504,
482476262400,
483548738400,
494122282290,
502612830720,
505159855200,
513480135168,
518453342208,
520212037632,
547929930240,
583096381560,
586207480320,
603567619200,
616719527424,
618269652000,
626112396000,
633926092800,
652482082560,
653289436800,
661576406400,
666574634880,
677701763200,
693688413600,
703816286208,
704575228896,
713178090240,
726673802400,
726972637440,
753132796416,
765181053000,
779729094144,
783990099200,
793104238080,
819730138500,
830350521000,
861743282400,
863638364160,
869516291840,
888875820360,
888988066400,
893835790848,
906550977024,
945884459520,
950432517216,
970956604800,
995024181060,
997978703400,
1018809792000,
1058501001600,
1070373679200,
1076349859200,
1085239701000,
1103539437000,
1109541413120,
1135890756000,
1144136294400,
1159571485800,
1161528261600,
1175104476000,
1179832600464,
1200229430400,
1209584724480,
1211621062400,
1219581548640,
1233377308800,
1253107608480,
1288623772800,
1324245491712,
1325481830400,
1330464844800,
1331785072800,
1369947647250,
1377031864320,
1386998613000,
1413817996500,
1438233280512,
1447428787200,
1480003190400,
1482760097280,
1507838492160,
1517389419000,
1542738616320,
1553357978368,
1556017837920,
1567241676000,
1571198926080,
1578475971072,
1584792261000,
1599300612000,
1626268644000,
1656012758400,
1681994012160,
1683038945280,
1708842189600,
1721209990500,
1773515487744,
1784852619264,
1801169758080,
1862961762816,
1886043571200,
1888271330400,
1919938116096,
1924339334400,
1948245082112,
1959868310400,
1987794251520,
2015156183040,
2020639420800,
2021976333000,
2033105289600,
2051203714560,
2059445329920,
2061489484800,
2066882988800,
2070303429600,
2096328767456,
2112394079250,
2128528765440,
2130069916800,
2172650274816,
2183877423000,
2198278051200,
2236152828000,
2259816300000,
2267834849280,
2312019021312,
2335483332000,
2363575441500,
2439654963200,
2448134325000,
2448278300160,
2468667064500,
2471771484000,
2520477679104,
2567400675840,
2608548875520,
2627456832000,
2644660418400,
2677752441000,
2706066874368,
2708593305600,
2708845856640,
2709493768800,
2827553208480,
2915401724928,
2965353955200,
3076882754400,
3105356994432,
3175969724928,
3218345676000,
3238966130400,
3321402084000,
3356538237000,
3377333836800,
3398177502720,
3448576989000,
3500961340800,
3519081431040,
3522876144480,
3531726240768,
3607776900000,
3622293071600,
3634863187200,
3772440804608,
3777406841600,
3881325763840,
3946161492000,
3962552630400,
3990762504960,
3991394534400,
4029093232640,
4205037804800,
4224973334400,
4240965560832,
4314435969536,
4346661822720,
4409499089268,
4437102673920,
4517245877760,
4537735429500,
4603679570880,
4612268729250,
4638285943200,
4660073935104,
4694568278400,
4712844296160,
4713692054400,
4741836503040,
4752162586080,
4824711643136,
4832764209000,
4903097162600,
4959751305600,
5085231579136,
5111051997870,
5148385482240,
5268640785408,
5289640356000,
5290460648928,
5431874152320,
5469709639680,
5681022328800,
5745853670400,
5808057260544,
5853911263200,
5914045683000,
5914410203520,
5956949980800,
6045468549120,
6073712944992,
6175225017000,
6200648966400,
6312101796000,
6343192620800,
6352588408320,
6355147895040,
6669629366400,
6734495875072,
6764077878600,
6793110213120,
6844445080704,
6884622108000.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 937 values, from 1 to 98079457512960).
n\r | 0 | 1 |
2 | 936 | 1 | 2 |
3 | 827 | 37 | 73 | 3 |
4 | 905 | 1 | 31 | 0 | 4 |
5 | 749 | 50 | 39 | 52 | 47 | 5 |
6 | 827 | 1 | 73 | 0 | 36 | 0 | 6 |
7 | 804 | 24 | 12 | 19 | 27 | 22 | 29 | 7 |
8 | 857 | 1 | 22 | 0 | 48 | 0 | 9 | 0 | 8 |
9 | 701 | 20 | 22 | 79 | 8 | 19 | 47 | 9 | 32 | 9 |
10 | 749 | 1 | 39 | 0 | 47 | 0 | 49 | 0 | 52 | 0 | 10 |
11 | 416 | 64 | 47 | 37 | 52 | 47 | 57 | 48 | 57 | 54 | 58 |
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.