Zumkeller numbers: Difference between revisions

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Tested odd zumkeller numbers up to 500,000,000 aka 5*10^8 , which takes 14h <BR>

'''All 205283 odd abundant numbers less than 10^8 that have even abundance are Zumkeller numbers. - T. D. Noe, Nov 14 2010'''<BR>

And yes there was no odd zumkeller with odd abundance->last prime factor must have a even power >0 2,4,6...

<pre> 1 h5m33 ->138978315 runtime 3933.063 s

+12 h ->482567085

+13 h ->501864363 stopped

count;number;SumOfDivs;CountofDivs;number : prime decomposition

10159 4999995; 12257280; 128 ; 4999995 : 3^3*5*7*11*13*37

10160 5000625; 10396672; 60 ; 5000625 : 3^2*5^4*7*127

20637 9999825; 20570112; 96 ; 9999825 : 3*5^2*11*17*23*31

20638 10000305; 20217600; 48 ; 10000305 : 3^2*5*7*53*599

41463 19999875; 40135680; 64 ; 19999875 : 3*5^3*7*19*401

41464 20000925; 45372096; 120 ; 20000925 : 3^4*5^2*7*17*83

62002 29999475; 66852864; 144 ; 29999475 : 3^2*5^2*11*17*23*31

62003 30000915; 62208000; 64 ; 30000915 : 3^3*5*7*53*599

82513 39999645; 82697472; 48 ; 39999645 : 3^2*5*7*23*5521

82514 40000275; 82985760; 72 ; 40000275 : 3^2*5^2*7*109*233

103001 49999635; 104884416; 48 ; 49999635 : 3^2*5*7*17*9337

103002 50000895; 102735360; 48 ; 50000895 : 3^5*5*7*5879

205283 99999375; 200935680; 80 ; 99999375 : 3*5^4*7*19*401 //205283 odd abundant numbers less than 10^8

205284 100000845; 217728000; 128 ; 100000845 : 3^3*5*7*29*41*89

307454 149999535; 307841040; 72 ; 149999535 : 3^2*5*7^2*59*1153

307455 150000165; 306748416; 48 ; 150000165 : 3^2*5*7*31*15361

409569 199999305; 400443264; 48 ; 199999305 : 3^2*5*11*17*23767

409570 200000115; 403269984; 36 ; 200000115 : 3^2*5*7^2*90703

511807 249999435; 521314560; 48 ; 249999435 : 3^2*5*7*19*41771

511808 250000065; 506782848; 48 ; 250000065 : 3^2*5*7*43*18457

613721 299998755; 609523200; 32 ; 299998755 : 3^3*5*7*317459

613722 300000645; 611696640; 64 ; 300000645 : 3^3*5*7*523*607

715573 349998705; 742072320; 64 ; 349998705 : 3^3*5*7*23*16103

715574 350000595; 717044064; 40 ; 350000595 : 3^4*5*7*123457

817456 399998865; 808206336; 64 ; 399998865 : 3*5*7*17*23*9743

817457 400000293; 803312640; 96 ; 400000293 : 3^2*7*11*17*19*1787

919451 449999235; 922184640; 48 ; 449999235 : 3^2*5*7*29*49261

919452 450000495; 943841280; 64 ; 450000495 : 3^3*5*7*31*15361

1021438 499999185; 1052101440; 72 ; 499999185 : 3^2*5*7^2*23*9859

1021439 500000445; 1027178880; 48 ; 500000445 : 3^5*5*7*58789

ODD Zumkeller Index;CoD Count of Divisors;Average CoD;Zumkeller number

1; 16; 16.0000;945 : 3^3*5*7

2; 18; 18.0000;1575 : 3^2*5^2*7

4; 20; 19.0000;2835 : 3^4*5*7

5; 24; 24.0000;3465 : 3^2*5*7*11

24; 32; 24.4211;10395 : 3^3*5*7*11

36; 36; 27.5000;17325 : 3^2*5^2*7*11

69; 40; 30.4242;31185 : 3^4*5*7*11

101; 48; 33.3125;45045 : 3^2*5*7*11*13

246; 54; 37.7931;121275 : 3^2*5^2*7^2*11

272; 64; 41.1538;135135 : 3^3*5*7*11*13

439; 72; 43.2335;225225 : 3^2*5^2*7*11*13

814; 80; 45.8027;405405 : 3^4*5*7*11*13

1366; 96; 49.0688;675675 : 3^3*5^2*7*11*13

3126; 108; 53.1364;1576575 : 3^2*5^2*7^2*11*13

4025; 120; 55.8843;2027025 : 3^4*5^2*7*11*13

4573; 128; 56.6752;2297295 : 3^3*5*7*11*13*17

7715; 144; 58.4978;3828825 : 3^2*5^2*7*11*13*17

14116; 160; 61.1239;6891885 : 3^4*5*7*11*13*17

23733; 192; 63.7696;11486475 : 3^3*5^2*7*11*13*17

55414; 216; 67.8331;26801775 : 3^2*5^2*7^2*11*13*17

71154; 240; 70.7454;34459425 : 3^4*5^2*7*11*13*17

89984; 256; 72.0990;43648605 : 3^3*5*7*11*13*17*19

149543; 288; 74.2782;72747675 : 3^2*5^2*7*11*13*17*19

268601; 320; 77.3901;130945815 : 3^4*5*7*11*13*17*19

446838; 384; 80.4003;218243025 : 3^3*5^2*7*11*13*17*19

last 501864363

1025232 84.3602;

1025232 80.7680;501864363 : 3^3*7*11*13*31*599

CntOfDivs|Count | last occurence

16 1 945

18 2 2205

20 1 2835

24 37 32445

28 1 25515

30 5 108045

32 44011 501862095

36 47777 501859575

40 16114 501848865

42 6 5294205

44 1 2066715

48 197324 501856425

50 4 972405

52 1 18600435

54 2795 499226175

56 2254 501650415

60 11478 501809175

64 154291 501863985

66 4 19190925

70 8 47647845

72 102440 501846975

78 4 172718325

80 34227 501843195

84 1892 501788925

88 71 498078315

90 1987 501185475

96 211623 501863175

98 4 428830605

100 670 500833125

104 8 496897335

108 13834 501858225

110 3 479773125

112 4445 501803505

120 20477 501833475

126 310 501701445

128 67263 501864363

132 32 485678025

140 138 500731875

144 48901 501862725

150 136 500788575

160 11220 501808125

162 454 501511725

168 1299 501570225

176 4 456744015

180 1599 501780825

192 19903 501838155

200 173 501744375

210 6 481340475

216 2097 501752475

224 274 501854535

240 1368 501860205

252 42 499200975

256 1163 501756255

270 21 495685575

280 1 456080625

288 871 501666165

300 2 463876875

320 92 499926735

324 9 471395925

336 6 485269785

360 13 500675175

384 32 498513015