Various Amicable Pair Lists and Statistics

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Lists of amicable pairs with members coprime to 30

Updated 28-Sep-2007

Lists of amicable pairs of type (i,1)

Updated 28-Sep-2007

Lists of amicable pairs with the same pair sum

Updated 28-Sep-2007

Lists of amicable pairs with pair sum not divisible by 9

Updated 28-Sep-2007

Discoverer overview

Updated 28-Sep-2007

Discoverer No of pairs
Pedersen8846325
Te Riele/Pedersen1343806
Garcia1245122
Ball165220
Costello92749
Walker&Einstein79457
Vom Stein&Borho74729
Borho&Battiato37785
Walker36502
Einstein17122
Garcia&Dubner13155
Wiethaus10401
Te Riele7783
Marcus5711
Einstein&Moews4247
Borho&Hoffmann3471
Jobling&Walker2872
Moews&Moews2614
Lee847
Zweers776
Kohmoto656
Woods471
Yuanhua465
Gubanov455
Chernych452
Moews295
Escott219
Poulet105
Needham93
Borho92
Knight72
Yan68
Cohen62
Euler59
David20
Marcus&Pedersen19
Ball/Costello14
Mason14
Bratley&McKay14
Alanen&Ore&Stemple8
Gerardin5
Ball&Jobling&Walker4
Wulf4
Gerardin&Poulet4
Battiato3
Garcia&Dubner&Jobling&Walker3
Seelhoff2
Nelson2
Dickson2
Yazdi/Decartes1
Yan&Jackson1
Rolf1
Pythagoras1
Paganini1
Legendre1
Brown1
al-Banna/Farisi/Fermat1
Wiethaus/Garcia1
Costello&Melvin1
Baader1
Total11412907

Milestones in exhaustive searches for amicable pairs

Limit Who When No of pairs below limit New pairs found in search
6232 Dickson 1913 5 0
15000 Brown 1939 7 1
105 Rolf 1965 13 1
106 Alanen&Ore&Stemple 1967 42 8
107 Bratley&McKay 1968 108 14
108 Cohen 1969 236 56
1010 Te Riele 1984 1427 816
2×1010 Te Riele 1990 1846 333
1011 Moews&Moews 1992 3340 1262
2×1011 Moews&Moews 1993 4310 860
3×1011 Moews&Moews 1996 4961 463
1012 Einstein&Moews 1996 7642 1965
Almost 1013 Einstein 2000 17509 8650
1013 Chernych 2002 17519 10
1014 Einstein&Walker 2003 39374 2050

Number of amicable pairs of each type

Updated 28-Sep-2007

Number of regular type (i,j) amicable pairs
j=1 j=2 j=3 j=4 j=5 j=6 j=7 j=8 total
i=2 506 4991 1315 259 34 2 - - 7107
i=3 1326 331791 36863 10640 610 19 1 - 381050
i=4 562 4552882 54194 26592 3128 74 1 1 4637434
i=5 152 2864886 14558 14032 4848 355 4 - 2898835
i=6 26 2010601 1545 2683 1853 323 7 - 2017038
i=7 8 1034871 63 189 229 75 1 - 1035436
i=8 - 977030 4 5 11 1 - - 977051
i=9 - 6177 - - - - - - 6177
total 2380 11783229 108542 54400 10713 849 14 1 11960128

Number of irregular type (i,j) amicable pairs
j=1 j=2 j=3 j=4 j=5 j=6 total
i=1 6 39 30 8 2 - 85
i=2 24 578 789 155 11 16 1573
i=3 61 1743 4584 1744 116 7 8255
i=4 24 3035 5060 4173 712 43 13047
i=5 1 2604 1654 2542 908 78 7787
i=6 1 1207 198 510 314 48 2278
i=7 - 1159 8 38 19 2 1226
i=8 - - 1 - - - 1
i=9 - 7 - - - - 7
total 117 10372 12324 9170 2082 194 34259

Number of amicable pairs of each type with smallest member below 1014

All pairs with smaller member below 1014 are known.

Number of regular type (i,j) amicable pairs
j=1 j=2 j=3 j=4 j=5 j=6 total
i=2 45 288 256 21 - - 610
i=3 66 2169 4731 1217 57 1 8241
i=4 16 1289 7257 5151 639 10 14362
i=5 4 190 2012 3449 1044 41 6740
i=6 - 16 203 462 256 21 958
i=7 - - 3 22 20 2 47
total 131 3952 14462 10322 2016 75 30958

Number of irregular type (i,j) amicable pairs
j=1 j=2 j=3 j=4 j=5 j=6 total
i=1 5 27 12 - - - 44
i=2 13 318 317 43 - - 691
i=3 36 728 1637 529 32 3 2965
i=4 5 377 1646 1202 171 5 3406
i=5 - 78 424 543 131 10 1186
i=6 1 6 24 58 28 3 120
i=7 - - - 3 1 - 4
total 60 1534 4060 2378 363 21 8416

Earliest examples of each amicable pair type

Earliest examples of regular type (i,j) amicable pairs
j=1 j=2 j=3 j=4 j=5 j=6 j=7 j=8
i=2 Phytagoras -500 Euler 1747 Euler 1747 Cohen 1969 Te Riele 1982 Gubanov 2006
i=3 Poulet 1941 Euler 1747 Euler 1747 David 1972 Moews&Moews 1992 Gubanov 2001 Gubanov 2006
i=4 Lee 1966 Poulet 1941 Mason 1921 Lee 1966 Lee 1966 Einstein 1997 Gubanov 2006 Gubanov 2006
i=5 Te Riele 1982 Te Riele 1982 David 1972 Te Riele 1990 Te Riele 1984 Einstein 1997 Walker&Einstein 2002
i=6 Pedersen 1997 Wiethaus 1988 Te Riele 1990 Te Riele 1990 Moews&Moews 1993 Einstein 1997 Walker&Einstein 2002
i=7 Garcia 2001 Wiethaus 1988 Einstein 1997 Einstein 1997 Einstein 1997 Einstein 1997 Walker&Einstein 2002
i=8 Pedersen 1999 Walker&Einstein 2002 Einstein 1997 Einstein 1997 Walker&Einstein 2002
i=9 Pedersen 2001

Earliest examples of irregular type (i,j) amicable pairs
j=1 j=2 j=3 j=4 j=5 j=6
i=1 Brown 1939 Paganini 1860 Moews&Moews 1993 Gubanov 2007 Gubanov 2007
i=2 Euler 1747 Euler 1747 Lee 1966 Te Riele 1985 Walker&Einstein 2002 Gubanov 2007
i=3 Poulet 1941 Poulet 1941 Poulet 1941 Te Riele 1984 Moews&Moews 1992 Einstein 1997
i=4 Cohen 1969 Te Riele 1984 Cohen 1969 David 1972 Te Riele 1990 Einstein 1998
i=5 Gubanov 2006 Te Riele 1984 Te Riele 1984 Te Riele 1984 Moews&Moews 1993 Einstein 1997
i=6 Einstein 1997 Einstein&Moews 1996 Einstein 1997 Te Riele 1990 Einstein 1997 Einstein 1997
i=7 Pedersen 2004 Einstein 1997 Einstein 1997 Einstein 1997 Walker&Einstein 2002
i=8 Walker&Einstein 2002
i=9 Pedersen 2007

Smallest known example of each amicable pair type

Pairs shown in italics are not necesarily the smallest pair of the type.

Regular pairs of type (i,j)
(2,1) Pythagoras -500 3D
220=2^2*5*11
284=2^2*71
(2,2) Euler 1747 4D
2620=2^2*5*131
2924=2^2*17*43
(2,3) Escott 1946 7D
1511930=2*5*7*21599
1598470=2*5*19*47*179
(2,4) Cohen 1969 8D
37784810=2*7*5*539783
39944086=2*7*13*41*53*101
(2,5) Gubanov 2006 15D
415620108287536=2^4*29*895732991999
417411574272464=2^4*59*127*373*449*20789
(2,6) Gubanov 2006 24D
434721341775145582079450=2*5^2*11*790402439591173785599
447367780808604362650150=2*5^2*29*107*113*223*541*211510704439
(3,1) Escott 1946 7D
6955216=2^4*19*137*167
7418864=2^4*463679
(3,2) Euler 1747 6D
100485=3^2*5*7*11*29
124155=3^2*5*31*89
(3,3) Lee 1966 7D
1077890=2*5*11*41*239
1099390=2*5*17*29*223
(3,4) TeRiele 1984 9D
155578180=2^2*5*223*34883
172610492=2^2*23*41*67*683
(3,5) Moews&Moews 1992 11D
75938508170=2*5*7*809*1340959
80471066230=2*5*23*67*101*149*347
(3,6) Walker&Einstein 2001 14D
35019687151575=3^2*5^2*7*4799*4633199
36680009488425=3^2*5^2*43*53*59*79*103*149
(3,7) Gubanov 2006 89D
74057091948435070376978994349560718661258033308183853288180780691316947007690396550194156=2^2*11*17*185999507590877374463*532296072715785415877368835700250207839880000021596034730231411519
75641200867118173487451825890641974844530027464943034881326246753049375788700028468829204=2^2*11*47*71*1061*84127*637808219*12789326141948728499383493*707556250730634490887449364146514124452707
(4,1) Lee 1966 10D
2099442345=3*5*7*11*13*37*3779
2533809495=3*5*7*24131519
(4,2) Lee 1966 7D
1669910=2*5*11*17*19*47
2062570=2*5*239*863
(4,3) Alanen&Ore&Stemple 1966 6D
667964=2^2*11*17*19*47
783556=2^2*31*71*89
(4,4) Cohen 1969 8D
32642324=2^2*11*13*149*383
35095276=2^2*17*47*79*139
(4,5) TeRiele 1984 9D/10D
996088412=2^2*11*17*151*8819
1030959268=2^2*23*37*41*83*89
(4,6) Einstein 2000 13D
1375168352756=2^2*11*19*179*9189599
1403766687244=2^2*23*29*41*59*199*1093
(4,7) Gubanov 2006 121D
2193151918713673169262080816226108719114907881679941441328538316157721206695821780538397395183124964766459280220624525550=2*5^2*11*8694426833447595610751*65618973925498519234003971057404710974929*6989331421513284568597790463019221536099879120377548419
2256952701803525479641125910490540930781439724625948417527130863788246755899798490026431957502258508463259047118842533650=2*5^2*29*107*113*223*541*211510704389*5044960319354424923906586262092721289772188857804888083405765767108440010644244792469313440601841
(4,8) Gubanov 2006 129D
296175653578845878211867966332139983090147867430303446399322873151696283755604883770555826880203072799028342058897106791106258850=2*5^2*11*8694427433573967532159*1151138556988832446809445777159504782348042614838859*53804435371060136537507721511427869146672997361590847
304791672592048667396282337239665624568023410724759511557939262154198192950879491078238770133769853293789603806198648430134777950=2*5^2*29*107*113*223*541*211510704389*129626254982904363953932248793*5255879745466143631755248980460348889342019361990958169897890403016551723871
(5,1) Garcia 1994 13D
3322710816776=2^3*13*31*37*1031*27017
3797383790584=2^3*474672973823
(5,2) TeRiele 1984 9D
227443340=2^2*5*17*43*47*331
302651764=2^2*1187*63743
(5,3) TeRiele 1984 9D
208693628=2^2*11*13*23*29*547
255308932=2^2*31*167*12329
(5,4) TeRiele 1984 10D
2310786764=2^2*11*13*19*149*1427
2727197236=2^2*29*79*139*2141
(5,5) TeRiele 1984 9D
161088158=2*7*11*17*37*1663
166706530=2*5*23*31*103*227
(5,6) Einstein 1997 13D
2355902938972=2^2*11*23*41*1109*51199
2456176165028=2^2*17*59*73*79*83*1279
(5,7) Walker&Einstein 2002 15D
186236493641692=2^2*13*17*79*6803*391999
190154254518308=2^2*29*31*41*59*97*179*1259
(6,1) Pedersen 2005 21D
456147017077210189227=3^5*7*17*31*19*23*41*53*743*721207
523639847195772482133=3^5*7*17*31*584141092577279
(6,2) Moews&Moews 1993 12D
104097591290=2*7*5*11*23*61*167*577
145582428166=2*7*2267*4587007
(6,3) TeRiele 1990 11D
12101753092=2^2*11*13*31*53*79*163
14559766268=2^2*23*359*440831
(6,4) TeRiele 1990 11D
10978923748=2^2*13*17*19*41*107*149
13025588252=2^2*23*139*971*1049
(6,5) Moews&Moews 1993 12D
162284213468=2^2*13*17*31*41*43*3359
188217223972=2^2*23*139*191*263*293
(6,6) Einstein 1997 13D
1977608096204=2^2*13*17*79*97*443*659
2075064774196=2^2*19*41*73*83*131*839
(6,7) Walker&Einstein 2002 15D
404450955224668=2^2*11*37*71*83*1063*39659
410195182795172=2^2*17*31*89*97*113*151*1321
(7,1) Pedersen 2002 48D
364319627806938632085663804753892835514458361075=3^5*5^2*41*11*101*199*4391*4919*4775729632963*64136506948868587
403565490769583860507694859170550795434188550925=3^5*5^2*41*1620256913658873273141402626399882747903999
(7,2) Pedersen 2003 31D
4331489803051762378810254684170=2*7*37*113*5*31*41*73*677*10247*22993510056263
5580389667027149572937888042998=2*7*37*113*8329069596671*11446160507207
(7,3) Einstein 1997 14D
24957782393030=2*5*19*23*41*83*89*109*173
27550372998970=2*5*17*156799*1033559
(7,4) Einstein 1997 13D
1361271112916=2^2*11*13*17*29*73*89*743
1785377488684=2^2*269*503*887*3719
(7,5) Einstein 1997 13D/14D
8819870180121=3^3*7*11*13*23*37*41*47*199
10948671003879=3^3*7*31*59*79*587*683
(7,6) Walker&Einstein 2001 14D
47585151910348=2^2*13*19*23*41*71*811*887
54984314191412=2^2*53*59*73*83*463*1567
(7,7) Walker&Einstein 2002 14D
4940469798236830=2*11*5*13*41*47*383*2087*2243
6028273830530402=2*11*23*31*97*101*173*197*1151
(8,2) Pedersen 2006 42D
484403941739617273345707212716447765367470=2*5*389*11*17*19*61*181*7001*15470057*29309098350923589743
602022790213134566335609133889438974702930=2*5*389*11032101028207035503*14028301801498682879
(8,3) Walker&Einstein 2002 16D
3599848247084570=2*5*13*23*29*43*59*73*97*2311
4431391475513830=2*5*167*1151*2305410799
(8,4) Einstein 1997 15D
226866310638188=2^2*13*17*19*43*59*89*163*367
279036019217812=2^2*31*307*2591*2828999
(8,5) Einstein 1997 15D/16D
814132108569364=2^2*13*17*19*23*139*157*269*359
1006370081510636=2^2*59*83*239*6299*34127
(8,6) Walker&Einstein 2002 16D
5809929463754636=2^2*11*19*29*31*59*311*601*701
6949168224309364=2^2*47*107*127*181*233*64499
(9,2) Pedersen 2005 94D
2040186716798418822898400998845135148225142117098584319982551190780724397977876299838471935924=2^2*19*11*23*37*71*619*94597*344405568041*150410344595981090857*13315572586250370474649445470153475683485688319
2422721726198122352191851186128597988517356262176019528755109344528895975381406800878682368076=2^2*19*5444804972860995629768363863299698978340126719*5854739996908449149318639127481181052814778879
Irregular pairs of type (i,j)
(1,1) Brown 1939 5D
12285=3^3*5*7*13
14595=3*5*7*139
(1,2) Paganini 1860 4D
1184=2^5*37
1210=2*5*11^2
(1,3) Moews&Moews 1993 12D
142742567625=3*5^3*7^2*17*456959
149814191415=3*5*7*17*181*509*911
(1,4) Gubanov 2007 16D
4880977214709375=3^4*5^5*13*1483297919
4933655063621505=3^4*5*13*53*89*8803*22567
(1,5) Gubanov 2007 34D
2216595640874012597754286043905125=3^3*5^3*13^3*298939043594667826194546239
2223007883359118222626159075582875=3^3*5^3*13*673*1019*1877*78203*503323188142829
(2,1) Euler 1750 4D
6232=2^3*19*41
6368=2^5*199
(2,2) Rolf 1964 5D
79750=2*5^3*11*29
88730=2*5*19*467
(2,3) Lee 1966 8D
10533296=2^4*19*34649
10949704=2^3*29*109*433
(2,4) Moews&Moews 1993 12D
133592967950=2*5^2*13*1109*185327
134246764210=2*5*13*73*109*233*557
(2,5) Walker&Einstein 2002 15D
684830058807050=2*5^2*13*995903*1057919
686939902296310=2*5*13*61*103*479*797*2203
(2,6) Gubanov 2007 66D
278602632047125828545781995625740498438248944298792781029040206625=3^3*5^3*13^3*54542695970281476274255515149*688881909793607334348029043263
279408583119451504841377946182970107345820823054200923046459313375=3^3*5^3*13*673*1019*1877*78203*503323188141989*125689425220441491803767944833429
(3,1) Cohen 1969 8D
46271745=3^2*5*13*19*23*181
49125375=3^2*5^3*13*3359
(3,2) Alanen&Ore&Stemple 1966 6D
280540=2^2*5*13^2*83
365084=2^2*107*853
(3,3) Bratley&McKay 1967 7D
5232010=2*7*5*41*1823
5799542=2*7^2*23*31*83
(3,4) TeRiele 1984 9D
359156770=2*7^2*5*83*8831
402020318=2*7*17*37*71*643
(3,5) Moews&Moews 1992 11D
31960479850=2*5^2*13*271*181439
32295407510=2*5*19*47*61*101*587
(3,6) Einstein 1997 13D
5851730304315=3^2*5*7^3*911*416159
5989852799685=3^2*5*23*29*37*67*79*1019
(4,1) Cohen 1969 8D
56512610=2*7*5*11*23*3191
75866014=2*7^2*774143
(4,2) TeRiele 1984 9D
133089500=2^2*5^3*23*71*163
176374564=2^2*383*115127
(4,3) Cohen 1969 8D
46237730=2*7*5*11^2*53*103
61319902=2*7*83*113*467
(4,4) David 1972 9D
318580262=2*7^2*11*13*127*179
343312858=2*7*17*23*59*1063
(4,5) TeRiele 1990 11D
13243828730=2*5*11^2*47*179*1301
13686947590=2*5*29*41*61*113*167
(4,6) Walker&Einstein 2002 14D
17555617000922=2*7^3*11*13^2*607*22679
18782315287078=2*7*23*29*37*83*179*3659
(5,1) Gubanov 2006 24D
577888159075686953414048=2^5*61*577*593*1695641*510269449
582438459540849842810752=2^7*4550300465162889396959
(5,2) TeRiele 1984 10D/11D
8376676490=2*7*5*13*19^2*43*593
11698280566=2*7*967*864107
(5,3) TeRiele 1984 10D
4999722525=3^2*5^2*7*17*29*47*137
6532396515=3^2*5*59*863*2851
(5,4) TeRiele 1984 10D
3000833115=3*5*7*17*19*23*3847
3382537125=3*5^3*11*47*73*239
(5,5) Moews&Moews 1993 12D
122944162418=2*7*11^2*17*23*419*443
134201314702=2*7*13*71*73*113*1259
(5,6) Einstein 1997 13D
1038054813075=3^2*7*5^2*41*71*373*607
1178875294317=3^2*7*23*37*43*61*83*101
(6,1) Einstein 1997 14D
21737963521274=2*7*13*11*37*79*107*149*233
24726933502726=2*7^2*13*19408895999
(6,2) Einstein&Moews 1996 12D
627701938990=2*7*5*17*37*41*463*751
815754577106=2*7^2*71*117239807
(6,3) Einstein 1997 13D
1238118114310=2*7*5*11^2*31*107*127*347
1710218599418=2*7*251*607*801791
(6,4) TeRiele 1990 11D
14840132450=2*31*5^2*11*13*23*41*71
21445545118=2*31*17*47*97*4463
(6,5) Einstein 1997 13D
1147108255436=2^2*11^2*13*29*59*127*839
1375439968564=2^2*31*151*199*419*881
(6,6) Walker&Einstein 2001 14D
27935608330604=2^2*13*23*47*509*761*1283
28398149949076=2^2*19^2*31*41*101*239*641
(7,2) Pedersen 2005 61D/62D
8507993766013829628521120595482873636214306915168127352328825=3^3*5^2*11*13^2*89*8707*1555766287*19457566159*289036425230363287991223559219
10164417331078469025224347613708878766051861926175783682551175=3^3*5^2*122144145952299824872712741939*123283812978829421781012684799
(7,3) Einstein 1997 15D
100610358639735=3*5*7*29*37*41*47*61*71*107
112127385273225=3*5^2*7*269*2591*306431
(7,4) Walker&Einstein 2001 14D
39010677666532=2^2*11*13^2*23*37*67*101*911
49670027308316=2^2*53*1291*1823*99551
(7,5) Einstein 1997 14D
74939168462158=2*7^3*11*13*23*41*67*107*113
95193839102642=2*7^2*101*179*223*479*503
(7,6) Walker&Einstein 2002 16D
2083878729755108=2^2*11*13^3*61*79*97*107*431
2450016682366492=2^2*47*59*83*271*433*22679
(8,3) Walker&Einstein 2002 16D/17D
7526776592305828=2^2*11^2*13*17*37*41*179*479*541
10007834494951772=2^2*839*819503*3638879
(9,2) Pedersen 2007 262D
3026659086897086365742891641906013454480904476726686719965380258574319789831351042050223057524131741163602835290019476068999751827617967256030524863091567303767880371100148535688163063222438136528033155462355882037357069978449860318096522490770246923472656230910=2*7*23*643*5*37*41^2*34721*10823576227*4518272828371*528049819204432744447160089464883*38834129393079419158046492659838590052780355080359993723*1349998517931450921613183992607034338965476905927038467991425759221494796885104862684336917044485535417727283353578528809254795685314997
3823461282661741715466953488694629963126172059990686760192022605625319498745694931312953685634610431371702599393875290986335644085376989768881622706942070097536153396971441574626743783472959632285664528366451490062244438617982136004324954278255610087009648285186=2*7*23*643*13679069636596747791153119238353474255024103010845434210298927439881917194873771674104093426464795341765430457277878288821*1349998517931426414166494475768686401137545999897649594593825075858522040050414048186749350497002639656512964921827660527812241290831871

Type system of amicable pairs

An amicable pair (M,N) = (gm,gn) pair with g = gcd(M,N) is said to have type (i,j) if m has i different prime factors not dividing g, and n has j different prime factors not dividing g. If m and n are squarefree, and gcd(g,m) = gcd(g,n) = 1 then the pair is called regular, otherwise it is called irregular.

Last update: 28-Sep-2007

Jan Munch Pedersen, amicable@post.cybercity.dk