--------------------------------------- CARMICHAEL NUMBERS AND LEHMER'S PROBLEM Jan Kristian Haugland --------------------------------------- My birthdate rocks! 31 x 7 x 73 = the 9th Carmichael number!! Here is a list with each Carmichael number n up to 1,000,000,000 together with phi(n), (n-1)/phi(n) and the factorization of n. A brief discussion of Lehmer's problem can be found at the bottom of the page. 561 320 7/4 3 x 11 x 17 1105 768 23/16 5 x 13 x 17 1729 1296 4/3 7 x 13 x 19 2465 1792 11/8 5 x 17 x 29 2821 2160 47/36 7 x 13 x 31 6601 5280 5/4 7 x 23 x 41 8911 7128 5/4 7 x 19 x 67 10585 8064 21/16 5 x 29 x 73 15841 12960 11/9 7 x 31 x 73 29341 25920 163/144 13 x 37 x 61 41041 28800 57/40 7 x 11 x 13 x 41 46657 41472 9/8 13 x 37 x 97 52633 44064 43/36 7 x 73 x 103 62745 32384 31/16 3 x 5 x 47 x 89 63973 46656 1777/1296 7 x 13 x 19 x 37 75361 57600 157/120 11 x 13 x 17 x 31 101101 72000 337/240 7 x 11 x 13 x 101 115921 103680 161/144 13 x 37 x 241 126217 93312 1753/1296 7 x 13 x 19 x 73 162401 148480 35/32 17 x 41 x 233 172081 129600 239/180 7 x 13 x 31 x 61 188461 139968 1745/1296 7 x 13 x 19 x 109 252601 240000 421/400 41 x 61 x 101 278545 200704 2487/1792 5 x 17 x 29 x 113 294409 279936 1363/1296 37 x 73 x 109 314821 285120 53/48 13 x 61 x 397 334153 308448 13/12 19 x 43 x 409 340561 278784 215/176 13 x 17 x 23 x 67 399001 378000 19/18 31 x 61 x 211 410041 391680 67/64 41 x 73 x 137 449065 326592 11/8 5 x 19 x 29 x 163 488881 466560 679/648 37 x 73 x 181 512461 486000 949/900 31 x 61 x 271 530881 483840 79/72 13 x 97 x 421 552721 460800 2303/1920 13 x 17 x 41 x 61 656601 392000 67/40 3 x 11 x 101 x 197 658801 518400 61/48 11 x 13 x 17 x 271 670033 513216 47/36 7 x 13 x 37 x 199 748657 559872 1733/1296 7 x 13 x 19 x 433 825265 497664 5731/3456 5 x 7 x 17 x 19 x 73 838201 648000 1397/1080 7 x 13 x 61 x 151 852841 720000 2369/2000 11 x 31 x 41 x 61 997633 746496 433/324 7 x 13 x 19 x 577 1024651 962280 115/108 19 x 199 x 271 1033669 793152 563/432 7 x 13 x 37 x 307 1050985 684288 1327/864 5 x 13 x 19 x 23 x 37 1082809 839808 557/432 7 x 13 x 73 x 163 1152271 1111320 1829/1764 43 x 127 x 211 1193221 1134000 947/900 31 x 61 x 631 1461241 1399680 451/432 37 x 73 x 541 1569457 1354752 519/448 17 x 19 x 43 x 113 1615681 1533312 255/242 23 x 199 x 353 1773289 1411344 2239/1782 7 x 19 x 67 x 199 1857241 1782000 469/450 31 x 181 x 331 1909001 1840000 83/80 41 x 101 x 461 2100901 1800000 7003/6000 11 x 31 x 61 x 101 2113921 1866240 367/324 19 x 31 x 37 x 97 2433601 2156544 325/288 17 x 37 x 53 x 73 2455921 2099520 379/324 13 x 19 x 61 x 163 2508013 2425488 699/676 53 x 79 x 599 2531845 1850688 197/144 5 x 19 x 29 x 919 2628073 2146176 529/432 7 x 37 x 73 x 139 2704801 2318400 7/6 11 x 29 x 61 x 139 3057601 2963520 65/63 43 x 211 x 337 3146221 2721600 2497/2160 13 x 31 x 37 x 211 3224065 2359296 2099/1536 5 x 13 x 193 x 257 3581761 3424512 205/196 29 x 113 x 1093 3664585 2794176 661/504 5 x 29 x 127 x 199 3828001 3750000 638/625 101 x 151 x 251 4335241 4218240 2779/2704 53 x 157 x 521 4463641 3499200 4133/3240 7 x 13 x 181 x 271 4767841 4105728 1505/1296 13 x 19 x 97 x 199 4903921 4233600 139/120 11 x 31 x 73 x 197 4909177 3825792 1663/1296 7 x 13 x 73 x 739 5031181 4390848 55/48 19 x 23 x 29 x 397 5049001 4860000 187/180 31 x 271 x 601 5148001 4992000 33/32 41 x 241 x 521 5310721 4665600 461/405 13 x 37 x 61 x 181 5444489 5224576 817/784 29 x 197 x 953 5481451 5265000 937/900 31 x 151 x 1171 5632705 4128768 1397/1024 5 x 13 x 193 x 449 5968873 5778864 911/882 43 x 127 x 1093 6049681 5241600 277/240 11 x 31 x 113 x 157 6054985 4672512 6469/4992 5 x 53 x 73 x 313 6189121 6048000 307/300 61 x 241 x 421 6313681 5114880 79/64 11 x 17 x 19 x 1777 6733693 6613488 2969/2916 109 x 163 x 379 6840001 5472000 5/4 7 x 17 x 229 x 251 6868261 6667920 1817/1764 43 x 211 x 757 7207201 6758400 273/256 17 x 353 x 1201 7519441 7296000 1649/1600 41 x 241 x 761 7995169 6257088 23/18 7 x 13 x 103 x 853 8134561 7838208 1345/1296 37 x 109 x 2017 8341201 7200000 2317/2000 11 x 31 x 61 x 401 8355841 7372800 17/15 13 x 41 x 61 x 257 8719309 7884864 18631/16848 19 x 37 x 79 x 157 8719921 6969600 1101/880 7 x 23 x 41 x 1321 8830801 7056720 1115/891 7 x 19 x 67 x 991 8927101 8164800 1417/1296 31 x 37 x 43 x 181 9439201 9234000 46/45 61 x 271 x 571 9494101 8712000 959/880 23 x 61 x 67 x 101 9582145 7248384 349/264 5 x 23 x 97 x 859 9585541 7873200 5917/4860 7 x 31 x 163 x 271 9613297 8636544 187/168 19 x 29 x 73 x 239 9890881 6912000 10303/7200 7 x 11 x 13 x 41 x 241 10024561 9828000 51/50 71 x 271 x 521 10267951 10062360 4445/4356 67 x 331 x 463 10402561 9031680 129/112 13 x 29 x 41 x 673 10606681 9797760 1403/1296 31 x 43 x 73 x 109 10837321 9360000 6947/6000 11 x 31 x 61 x 521 10877581 8467200 8633/6720 11 x 13 x 29 x 43 x 61 11119105 8323072 171/128 5 x 17 x 257 x 509 11205601 9273600 29/24 11 x 17 x 31 x 1933 11921001 7629440 25/16 3 x 29 x 263 x 521 11972017 11648448 37/36 43 x 433 x 643 12261061 11188800 263/240 19 x 61 x 71 x 149 12262321 11059200 5677/5120 17 x 41 x 73 x 241 12490201 11337408 6425/5832 19 x 37 x 109 x 163 12945745 9144576 1427/1008 5 x 19 x 29 x 37 x 127 13187665 9834496 2403/1792 5 x 17 x 113 x 1373 13696033 11483136 229/192 13 x 17 x 29 x 2137 13992265 8895744 151/96 5 x 7 x 19 x 53 x 397 14469841 14206752 55/54 73 x 379 x 523 14676481 14406000 1248/1225 71 x 421 x 491 14913991 14447160 607/588 43 x 127 x 2731 15247621 14904000 3683/3600 61 x 181 x 1381 15403285 11883456 477/368 5 x 37 x 139 x 599 15829633 15410304 151/147 43 x 547 x 673 15888313 12702096 743/594 7 x 19 x 67 x 1783 16046641 14183424 1955/1728 13 x 37 x 73 x 457 16778881 12441600 971/720 7 x 17 x 19 x 41 x 181 17098369 16859136 6361/6272 113 x 337 x 449 17236801 17010000 76/75 151 x 211 x 541 17316001 16872960 2775/2704 53 x 157 x 2081 17586361 15681600 4441/3960 13 x 61 x 67 x 331 17812081 14601600 1903/1560 7 x 41 x 53 x 1171 18162001 13824000 1009/768 11 x 13 x 17 x 31 x 241 18307381 17035200 3353/3120 29 x 61 x 79 x 131 18900973 14872896 61/48 7 x 13 x 229 x 907 19384289 19081216 7867/7744 89 x 353 x 617 19683001 17496000 9/8 13 x 37 x 151 x 271 20964961 18282240 211/184 17 x 19 x 47 x 1381 21584305 16156672 171/128 5 x 17 x 197 x 1289 22665505 16883712 1031/768 5 x 17 x 97 x 2749 23382529 23003136 1171/1152 97 x 193 x 1249 25603201 22579200 127/112 13 x 29 x 113 x 601 26280073 25743744 49/48 73 x 157 x 2293 26474581 21170160 4457/3564 7 x 19 x 67 x 2971 26719701 17100608 25/16 3 x 29 x 197 x 1559 26921089 23887872 23369/20736 13 x 37 x 97 x 577 26932081 23699200 2091/1840 11 x 47 x 113 x 461 27062101 23400000 2313/2000 11 x 31 x 61 x 1301 27336673 20072448 8629/6336 7 x 13 x 23 x 37 x 353 27402481 25401600 5437/5040 31 x 43 x 61 x 337 28787185 18413568 1801/1152 5 x 7 x 19 x 73 x 593 29020321 25401600 2879/2520 11 x 37 x 113 x 631 29111881 28812000 4951/4900 211 x 281 x 491 31146661 23328000 57679/43200 7 x 13 x 31 x 61 x 181 31405501 30870000 997/980 71 x 631 x 701 31692805 24628032 803/624 5 x 47 x 157 x 859 32914441 26282880 541/432 7 x 19 x 61 x 4057 33302401 28800000 3469/3000 11 x 31 x 61 x 1601 33596641 29030400 1111/960 13 x 17 x 281 x 541 34196401 31159296 1475/1344 17 x 47 x 127 x 337 34657141 34314000 101/100 191 x 421 x 431 34901461 28226880 17627/14256 7 x 19 x 397 x 661 35571601 31233600 41/36 13 x 31 x 61 x 1447 35703361 34974720 49/48 61 x 277 x 2113 36121345 24772608 6719/4608 5 x 13 x 17 x 97 x 337 36765901 35691840 445/432 37 x 613 x 1621 37167361 27648000 9679/7200 7 x 11 x 41 x 61 x 193 37280881 31449600 569/480 11 x 17 x 73 x 2731 37354465 28449792 2647/2016 5 x 29 x 73 x 3529 37964809 37476432 1477/1458 109 x 379 x 919 38151361 35942400 1019/960 41 x 53 x 97 x 181 38624041 36391680 917/864 37 x 61 x 109 x 157 38637361 32037120 521/432 7 x 37 x 241 x 619 39353665 28901376 4183/3072 5 x 13 x 193 x 3137 40160737 35126784 1811/1584 19 x 23 x 29 x 3169 40280065 26763264 149/99 5 x 7 x 67 x 89 x 193 40430401 36000000 8423/7500 11 x 101 x 151 x 241 40622401 36691200 31/28 17 x 43 x 61 x 911 40917241 37149840 1249/1134 19 x 31 x 127 x 547 41298985 25913088 51/32 5 x 7 x 13 x 139 x 653 41341321 31492800 38279/29160 7 x 19 x 31 x 37 x 271 41471521 31104000 86399/64800 7 x 13 x 31 x 61 x 241 42490801 39600000 1073/1000 31 x 41 x 101 x 331 43286881 37440000 6937/6000 11 x 31 x 61 x 2081 43331401 42230160 905/882 43 x 631 x 1597 43584481 39398400 177/160 17 x 31 x 191 x 433 43620409 35178624 1607/1296 7 x 19 x 157 x 2089 44238481 37065600 1289/1080 7 x 61 x 313 x 331 45318561 28992320 941/602 3 x 29 x 173 x 3011 45877861 42865200 12137/11340 31 x 43 x 127 x 271 45890209 42771456 103/96 29 x 53 x 73 x 409 46483633 37604736 89/72 7 x 19 x 373 x 937 47006785 30108672 2473/1584 5 x 7 x 17 x 199 x 397 48321001 44928000 413/384 37 x 41 x 53 x 601 48628801 42768000 307/270 13 x 31 x 67 x 1801 49333201 45158400 839/768 17 x 61 x 113 x 421 50201089 49545216 4669/4608 97 x 673 x 769 53245921 48222720 53/48 17 x 41 x 79 x 967 53711113 53149824 24593/24336 157 x 313 x 1093 54767881 45209664 785/648 7 x 37 x 103 x 2053 55462177 49299712 9/8 17 x 23 x 83 x 1709 56052361 55566000 22243/22050 211 x 421 x 631 58489201 52920000 2321/2100 19 x 29 x 101 x 1051 60112885 41116608 1193/816 5 x 7 x 443 x 3877 60957361 59616000 409/400 61 x 181 x 5521 62756641 61896960 73/72 109 x 241 x 2389 64377991 63772920 8831/8748 163 x 487 x 811 64774081 60963840 17/16 29 x 71 x 163 x 193 65037817 56080512 167/144 13 x 19 x 73 x 3607 65241793 60850944 1471/1372 29 x 43 x 113 x 463 67371265 47775744 361/256 5 x 13 x 37 x 109 x 257 67653433 61471872 12203/11088 19 x 29 x 199 x 617 67902031 66066840 37/36 43 x 271 x 5827 67994641 54432000 1499/1200 11 x 13 x 37 x 71 x 181 68154001 67500000 11359/11250 151 x 601 x 751 69331969 53747712 7523/5832 7 x 19 x 37 x 73 x 193 70561921 65894400 257/240 31 x 53 x 67 x 641 72108421 55882800 1471/1140 7 x 11 x 191 x 4903 72286501 60231600 1685/1404 7 x 67 x 79 x 1951 74165065 52962816 2599/1856 5 x 13 x 59 x 83 x 233 75151441 63221760 14911/12544 17 x 19 x 29 x 71 x 113 75681541 65085120 1507/1296 13 x 19 x 61 x 5023 75765313 66576384 437/384 13 x 29 x 73 x 2753 76595761 62208000 35461/28800 11 x 17 x 31 x 73 x 181 77826001 68544000 109/96 11 x 43 x 137 x 1201 78091201 72705600 29/27 31 x 37 x 103 x 661 78120001 72316800 175/162 19 x 73 x 151 x 373 79411201 78643200 517/512 193 x 257 x 1601 79624621 78842160 2137/2116 139 x 691 x 829 80282161 76204800 15929/15120 43 x 61 x 127 x 241 80927821 70061760 499/432 13 x 19 x 103 x 3181 81638401 73728000 1063/960 13 x 97 x 101 x 641 81926461 75742128 1285/1188 19 x 67 x 139 x 463 82929001 82320000 3949/3920 281 x 421 x 701 83099521 74789568 10/9 19 x 37 x 43 x 2749 83966401 79027200 17/16 41 x 43 x 97 x 491 84311569 77780736 1717/1584 19 x 73 x 89 x 683 84350561 83371120 2842/2809 107 x 743 x 1061 84417985 65028096 997/768 5 x 29 x 577 x 1009 87318001 79833600 35/32 17 x 71 x 73 x 991 88689601 62208000 2053/1440 7 x 11 x 13 x 41 x 2161 90698401 89613120 585/578 103 x 647 x 1361 92625121 91854000 2042/2025 181 x 631 x 811 93030145 68419584 4177/3072 5 x 13 x 257 x 5569 93614521 65664000 41059/28800 7 x 11 x 13 x 41 x 2281 93869665 67436544 139687/100352 5 x 17 x 29 x 113 x 337 94536001 76574160 100/81 7 x 19 x 607 x 1171 96895441 95738112 3935/3888 109 x 433 x 2053 99036001 97200000 917/900 61 x 541 x 3001 99830641 95397120 653/624 53 x 79 x 113 x 211 99861985 78299136 653/512 5 x 97 x 109 x 1889 100427041 78382080 41/32 11 x 13 x 17 x 109 x 379 101649241 99792000 3667/3600 61 x 661 x 2521 101957401 75582720 94405/69984 7 x 13 x 19 x 109 x 541 102090781 84680640 5729/4752 13 x 19 x 31 x 67 x 199 104404861 91022400 1927/1680 11 x 37 x 71 x 3613 104569501 102202800 2165/2116 47 x 1151 x 1933 104852881 86605200 1471/1215 7 x 31 x 271 x 1783 105117481 80870400 7487/5760 7 x 19 x 37 x 41 x 521 105309289 99937152 4097/3888 37 x 73 x 127 x 307 105869401 91728000 277/240 11 x 29 x 79 x 4201 106041937 94228992 15449/13728 17 x 23 x 79 x 3433 107714881 102211200 137/130 37 x 71 x 131 x 313 109393201 103680000 10129/9600 37 x 73 x 101 x 401 109577161 101616768 138355/128304 19 x 73 x 199 x 397 111291181 103062960 1853/1716 23 x 43 x 131 x 859 114910489 113749776 99/98 127 x 659 x 1373 115039081 113892480 24581/24336 157 x 313 x 2341 115542505 81962496 6947/4928 5 x 13 x 43 x 67 x 617 116682721 115906560 451/448 281 x 617 x 673 118901521 118098000 18349/18225 271 x 541 x 811 119327041 114307200 1973/1890 61 x 73 x 127 x 211 120981601 107495424 46675/41472 13 x 37 x 97 x 2593 121247281 111767040 243/224 17 x 89 x 127 x 631 122785741 109771200 2953/2640 13 x 61 x 67 x 2311 124630273 122923008 73/72 109 x 229 x 4993 127664461 125538000 4983/4900 71 x 421 x 4271 128697361 111513600 48749/42240 13 x 17 x 661 x 881 129255841 101606400 12823/10080 11 x 13 x 19 x 113 x 421 129762001 118098000 89/81 19 x 31 x 151 x 1459 130032865 95883264 2951/2176 5 x 19 x 97 x 103 x 137 130497361 122068800 449/420 29 x 61 x 71 x 1039 132511681 121927680 313/288 19 x 43 x 241 x 673 133205761 110592000 11563/9600 13 x 17 x 41 x 61 x 241 133344793 123451776 9799/9072 19 x 73 x 127 x 757 133800661 132269760 437/432 109 x 541 x 2269 134809921 125411328 3715/3456 19 x 97 x 193 x 379 134857801 111767040 695/576 13 x 19 x 29 x 67 x 281 135556345 104315904 499/384 5 x 37 x 89 x 8233 136625941 124711488 38335/34992 19 x 37 x 109 x 1783 139592101 121950000 1717/1500 11 x 31 x 151 x 2711 139952671 138567000 101/100 131 x 571 x 1871 140241361 118540800 83477/70560 13 x 29 x 41 x 43 x 211 144218341 132269760 38153/34992 19 x 37 x 271 x 757 145124785 103638528 941/672 5 x 13 x 43 x 137 x 379 146843929 145496736 109/108 163 x 379 x 2377 150846961 142560000 57139/54000 31 x 61 x 241 x 331 151530401 125440000 27059/22400 11 x 17 x 71 x 101 x 113 151813201 144000000 126511/120000 41 x 61 x 101 x 601 153589801 152250000 1261/1250 151 x 701 x 1451 153927961 136086400 3891/3440 11 x 47 x 173 x 1721 157731841 149361408 3260/3087 29 x 113 x 127 x 379 158404141 125971200 293341/233280 7 x 31 x 37 x 109 x 181 158864833 146313216 39401/36288 19 x 43 x 337 x 577 159492061 148599360 1391/1296 19 x 127 x 157 x 421 161035057 152635392 12289/11648 29 x 113 x 157 x 313 161242705 111476736 53321/36864 5 x 13 x 17 x 337 x 433 161913961 134784000 34597/28800 11 x 31 x 37 x 41 x 313 163954561 160997760 1109/1089 67 x 331 x 7393 167979421 158760000 14813/14000 31 x 61 x 211 x 421 168659569 151449408 49/44 13 x 67 x 83 x 2333 169057801 139968000 31307/25920 11 x 19 x 41 x 109 x 181 169570801 142248960 11215/9408 17 x 19 x 29 x 43 x 421 170947105 117024192 149/102 5 x 7 x 647 x 7549 171454321 170061120 26459/26244 163 x 811 x 1297 171679561 164511360 110051/105456 53 x 79 x 131 x 313 172290241 162155520 17/16 37 x 41 x 137 x 829 172430401 138240000 35923/28800 11 x 13 x 31 x 97 x 401 172947529 171915696 47099/46818 307 x 613 x 919 173085121 146016000 1541/1300 11 x 31 x 53 x 61 x 157 174352641 114329600 61/40 3 x 71 x 641 x 1277 175997185 111034368 913/576 5 x 7 x 13 x 503 x 769 176659201 165888000 3067/2880 37 x 41 x 101 x 1153 178451857 166281984 37553/34992 19 x 109 x 199 x 433 178482151 176715000 101/100 151 x 331 x 3571 178837201 175600800 1713/1682 59 x 1451 x 2089 180115489 178564176 2206/2187 163 x 487 x 2269 181154701 138205440 755/576 7 x 13 x 37 x 173 x 311 182356993 140341248 421/324 7 x 13 x 73 x 97 x 283 184353001 152568000 29/24 7 x 31 x 653 x 1301 186393481 184548000 101/100 131 x 521 x 2731 186782401 167215104 46325/41472 13 x 37 x 577 x 673 187188001 131328000 821/576 7 x 11 x 13 x 41 x 4561 188516329 186041856 7847/7744 89 x 617 x 3433 188689501 135000000 13977/10000 7 x 11 x 13 x 251 x 751 189941761 188743680 2061/2048 257 x 641 x 1153 193708801 152928000 19/15 11 x 13 x 31 x 37 x 1181 193910977 176569344 123/112 13 x 139 x 239 x 449 194120389 185177664 85591/81648 37 x 109 x 127 x 379 194675041 166060800 4361/3720 11 x 17 x 373 x 2791 196358977 180652032 30991/28512 19 x 67 x 73 x 2113 200753281 197568000 4979/4900 71 x 631 x 4481 206955841 191036160 13/12 17 x 71 x 277 x 619 208969201 169361280 8795/7128 7 x 19 x 661 x 2377 212027401 203546304 25/24 53 x 83 x 157 x 307 213835861 191464560 2533/2268 19 x 31 x 43 x 8443 214850881 181398528 1535/1296 7 x 109 x 193 x 1459 214852609 213342336 3272/3249 229 x 457 x 2053 216821881 215622000 18251/18150 331 x 661 x 991 221884001 219700000 4267/4225 131 x 521 x 3251 225745345 147197952 3239/2112 5 x 7 x 23 x 193 x 1453 226509361 205083648 3605/3264 13 x 89 x 137 x 1429 227752993 206032896 11461/10368 13 x 97 x 109 x 1657 228842209 227598336 113513/112896 337 x 673 x 1009 230630401 221405184 25/24 53 x 97 x 113 x 397 230996949 150582848 1939/1264 3 x 53 x 317 x 4583 231194965 176121792 13169/10032 5 x 23 x 229 x 8779 237597361 223891200 3311/3120 31 x 47 x 313 x 521 238244041 200275200 571/480 7 x 73 x 191 x 2441 238527745 187858944 3413/2688 5 x 113 x 157 x 2689 241242001 172800000 40207/28800 7 x 11 x 13 x 401 x 601 242641153 207028224 8101/6912 17 x 19 x 37 x 79 x 257 246446929 229267584 6269/5832 19 x 109 x 127 x 937 247095361 226748160 9533/8748 19 x 37 x 271 x 1297 250200721 228614400 49643/45360 19 x 31 x 421 x 1009 252141121 239500800 379/360 43 x 61 x 97 x 991 255160621 242049600 253/240 37 x 61 x 131 x 863 256828321 243855360 2831/2688 29 x 113 x 181 x 433 257495641 229262400 1213/1080 13 x 31 x 367 x 1741 258634741 256074000 101/100 131 x 491 x 4021 266003101 203233536 671725/513216 7 x 13 x 37 x 199 x 397 270857521 223948800 13933/11520 11 x 19 x 41 x 73 x 433 271481329 219625344 89/72 7 x 19 x 277 x 7369 271794601 237202560 55/48 13 x 19 x 743 x 1481 273769921 254361600 4133/3840 17 x 181 x 193 x 461 274569601 252334080 3865/3552 17 x 61 x 149 x 1777 275283401 270970000 2489/2450 71 x 701 x 5531 277241401 262440000 51341/48600 31 x 61 x 271 x 541 278152381 242751168 55/48 13 x 19 x 727 x 1549 279377281 276822000 328/325 131 x 911 x 2341 280067761 261273600 18523/17280 29 x 41 x 109 x 2161 280761481 196992000 41047/28800 7 x 11 x 13 x 41 x 6841 288120421 286526160 94157/93636 307 x 613 x 1531 289766701 233979840 535/432 7 x 19 x 181 x 12037 289860481 275365888 46215/43904 29 x 113 x 197 x 449 291848401 252979200 443/384 13 x 17 x 541 x 2441 292244833 267898752 3181/2916 19 x 37 x 199 x 2089 292776121 244944000 4303/3600 11 x 31 x 41 x 43 x 487 295643089 250765632 863/732 7 x 157 x 367 x 733 295826581 283435200 182609/174960 37 x 163 x 181 x 271 296559361 238878720 8581/6912 7 x 31 x 73 x 97 x 193 299736181 297432000 32651/32400 181 x 541 x 3061 300614161 279417600 16267/15120 31 x 43 x 61 x 3697 301704985 227017728 1531/1152 5 x 43 x 47 x 73 x 409 302751505 203751936 123673/83232 5 x 7 x 103 x 137 x 613 306871201 233280000 14207/10800 7 x 13 x 31 x 181 x 601 311388337 260112384 44131/36864 13 x 17 x 37 x 113 x 337 318266641 238464000 19219/14400 7 x 11 x 41 x 73 x 1381 321197185 217147392 71/48 5 x 19 x 23 x 29 x 37 x 137 321602401 302400000 2127/2000 31 x 41 x 401 x 631 325546585 205383168 913/576 5 x 7 x 13 x 359 x 1993 328573477 326020464 39511/39204 199 x 397 x 4159 329769721 323568000 1223/1200 61 x 421 x 12841 333065305 252502272 3419/2592 5 x 37 x 83 x 109 x 199 333229141 277369920 173/144 7 x 37 x 739 x 1741 334783585 252294912 16847/12696 5 x 37 x 47 x 139 x 277 338740417 314302464 2897/2688 19 x 59 x 449 x 673 346808881 301086720 43789/38016 19 x 23 x 37 x 89 x 241 348612265 266378112 691/528 5 x 23 x 859 x 3529 354938221 292481280 233/192 11 x 19 x 29 x 157 x 373 357277921 255744000 20117/14400 7 x 11 x 13 x 241 x 1481 357380101 311895360 55/48 13 x 19 x 631 x 2293 358940737 306561024 8093/6912 17 x 23 x 37 x 43 x 577 360067201 311040000 37507/32400 13 x 31 x 61 x 97 x 151 362569201 358256640 12695/12544 113 x 337 x 9521 364590721 298598400 7033/5760 11 x 17 x 31 x 109 x 577 366532321 363847680 2321/2304 241 x 337 x 4513 366652201 365010000 26569/26450 461 x 691 x 1151 367804801 279936000 473/360 7 x 13 x 31 x 241 x 541 367939585 254803968 1109/768 5 x 13 x 17 x 433 x 769 368113411 358956360 201/196 43 x 631 x 13567 382304161 379676640 291/289 239 x 409 x 3911 382536001 362250000 132/125 31 x 71 x 151 x 1151 390489121 326592000 12913/10800 11 x 19 x 61 x 109 x 281 392099401 373262400 1353/1288 29 x 139 x 211 x 461 393122521 317882880 2137/1728 11 x 19 x 29 x 37 x 1753 393513121 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407808000 4281/3200 7 x 11 x 41 x 61 x 2833 547652161 441112320 293/236 7 x 17 x 709 x 6491 548871961 470400000 653419/560000 11 x 41 x 61 x 71 x 281 549117205 344169216 919/576 5 x 7 x 13 x 103 x 11717 549333121 505774080 5839/5376 17 x 61 x 197 x 2689 549538081 519321600 7111/6720 29 x 61 x 241 x 1289 551672221 548171280 13079/12996 229 x 571 x 4219 552894301 525672000 589/560 43 x 61 x 101 x 2087 555465601 552960000 6429/6400 401 x 641 x 2161 556199281 471744000 8489/7200 11 x 37 x 43 x 61 x 521 556450777 518301504 16001/14904 19 x 109 x 139 x 1933 557160241 488880000 3419/3000 11 x 31 x 421 x 3881 557795161 421986240 2337/1768 7 x 11 x 53 x 103 x 1327 558570961 500152320 11579/10368 13 x 61 x 73 x 9649 558977761 551992320 3921/3872 89 x 881 x 7129 561481921 393984000 10261/7200 7 x 11 x 13 x 41 x 13681 561777121 497664000 14449/12800 17 x 41 x 61 x 73 x 181 564651361 551214720 1807/1764 43 x 3361 x 3907 568227241 508032000 225487/201600 31 x 37 x 41 x 43 x 281 569332177 566213424 14239/14161 239 x 1429 x 1667 573896881 535377600 15629/14580 19 x 109 x 181 x 1531 577240273 512096256 572659/508032 19 x 29 x 73 x 113 x 127 579606301 573168960 11795/11664 109 x 541 x 9829 580565233 559338624 18243/17576 53 x 79 x 313 x 443 590754385 429048576 727/528 5 x 13 x 109 x 199 x 419 593234929 455393664 1261/968 7 x 19 x 23 x 89 x 2179 595405201 576000000 496171/480000 61 x 101 x 241 x 401 597717121 588588000 1244/1225 71 x 911 x 9241 600892993 558047232 3997/3712 19 x 59 x 577 x 929 602074585 407503872 491891/332928 5 x 7 x 137 x 307 x 409 602426161 551577600 699/640 17 x 41 x 421 x 2053 602593441 544204800 1063/960 17 x 41 x 61 x 14173 606057985 438939648 6263/4536 5 x 13 x 109 x 113 x 757 609865201 518400000 18823/16000 11 x 31 x 41 x 181 x 241 611397865 402361344 110281/72576 5 x 13 x 19 x 29 x 43 x 397 612347905 404619264 339/224 5 x 13 x 17 x 29 x 97 x 197 612816751 609375000 62853/62500 251 x 751 x 3251 616463809 523763712 86777/73728 13 x 17 x 97 x 149 x 193 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x 401 697906561 623001600 27961/24960 11 x 97 x 131 x 4993 698548201 592704000 166321/141120 13 x 29 x 41 x 43 x 1051 702683101 699720000 19683/19600 421 x 701 x 2381 703995733 558892224 592589/470448 7 x 19 x 67 x 199 x 397 704934361 673920000 50209/48000 41 x 61 x 521 x 541 705101761 525533184 12365/9216 7 x 13 x 19 x 193 x 2113 707926801 641249280 5935/5376 17 x 29 x 337 x 4261 710382401 704327680 18655/18496 137 x 953 x 5441 710541481 642945600 13129/11880 13 x 61 x 331 x 2707 711374401 663552000 5489/5120 17 x 241 x 401 x 433 713588401 685843200 9439/9072 37 x 151 x 337 x 379 717164449 646918272 439/396 23 x 43 x 67 x 79 x 137 721244161 597196800 3913/3240 11 x 19 x 41 x 73 x 1153 722923201 518400000 50203/36000 7 x 11 x 13 x 401 x 1801 727083001 700131600 19235/18522 43 x 127 x 211 x 631 739444021 735860160 76547/76176 277 x 1381 x 1933 743404663 735417144 10517/10404 103 x 1123 x 6427 744866305 539492352 69277/50176 5 x 17 x 29 x 449 x 673 745864945 501997056 523/352 5 x 7 x 89 x 149 x 1607 746706961 534954240 67/48 7 x 11 x 13 x 277 x 2693 752102401 722534400 1399/1344 41 x 97 x 281 x 673 759472561 711244800 21527/20160 29 x 41 x 181 x 3529 765245881 666792000 11247/9800 19 x 29 x 31 x 71 x 631 771043201 768000000 80317/80000 401 x 1201 x 1601 775368901 772182720 138955/138384 373 x 1117 x 1861 775866001 748339200 3185/3072 41 x 233 x 241 x 337 776176261 653313600 182201/153360 7 x 61 x 853 x 2131 781347841 732513600 16/15 29 x 61 x 79 x 5591 784966297 777275904 18679/18496 137 x 409 x 14009 790020001 755048448 1875/1792 29 x 229 x 337 x 353 790623289 760900608 133/128 37 x 233 x 293 x 313 794937601 785645568 9325/9216 97 x 769 x 10657 798770161 674179200 2417/2040 7 x 73 x 1021 x 1531 804978721 797247360 6143/6084 157 x 313 x 16381 809702401 754974720 52715/49152 17 x 241 x 257 x 769 809883361 682007040 19/16 7 x 97 x 139 x 8581 814056001 713250000 428/375 11 x 31 x 251 x 9511 822531841 817889280 16477/16384 257 x 641 x 4993 824389441 810432000 1831/1800 61 x 3361 x 4021 824405041 748828800 1189/1080 19 x 31 x 109 x 12841 829678141 787320000 512147/486000 31 x 61 x 541 x 811 832060801 622080000 28891/21600 11 x 13 x 17 x 31 x 61 x 181 833608321 635904000 18877/14400 7 x 13 x 31 x 461 x 641 834244501 805942080 604525/584016 47 x 139 x 277 x 461 834720601 635040000 198743/151200 11 x 13 x 29 x 31 x 43 x 151 836515681 780595200 22633/21120 23 x 71 x 97 x 5281 839275921 711244800 23789/20160 11 x 29 x 37 x 211 x 337 841340521 700306560 173/144 7 x 37 x 571 x 5689 842202361 590976000 13681/9600 7 x 11 x 13 x 41 x 20521 843704401 801504000 2779/2640 31 x 67 x 401 x 1013 847491361 739584000 16501/14400 17 x 31 x 41 x 61 x 643 849064321 719207424 85/72 7 x 157 x 193 x 4003 851703301 845962800 31899/31684 179 x 1069 x 4451 851934601 699840000 473297/388800 7 x 61 x 73 x 151 x 181 852729121 797817600 3463/3240 31 x 37 x 163 x 4561 854197345 616771584 20741/14976 5 x 19 x 23 x 313 x 1249 855734401 849100800 129/128 193 x 401 x 11057 860056705 672841728 2127/1664 5 x 53 x 313 x 10369 863984881 859578480 7843/7803 307 x 613 x 4591 867800701 788486400 74171/67392 31 x 37 x 61 x 79 x 157 868234081 645120000 258403/192000 11 x 13 x 17 x 31 x 41 x 281 876850801 870534000 2221/2205 211 x 421 x 9871 882796321 762048000 9731/8400 11 x 43 x 71 x 97 x 271 885336481 846526464 16265/15552 37 x 73 x 433 x 757 888700681 792892800 269/240 11 x 71 x 229 x 4969 891706861 884912688 39505/39204 199 x 397 x 11287 897880321 784287360 3802/3321 13 x 19 x 1231 x 2953 902645857 883032192 2163/2116 47 x 3727 x 5153 914801665 693633024 42541/32256 5 x 29 x 97 x 193 x 337 918661501 911250000 7561/7500 151 x 751 x 8101 928482241 885104640 138167/131712 29 x 113 x 421 x 673 930745621 834442560 3373/3024 13 x 43 x 181 x 9199 931694401 778752000 4977/4160 11 x 17 x 73 x 131 x 521 934784929 804722688 14051/12096 13 x 23 x 73 x 113 x 379 935794081 921922560 877/864 73 x 937 x 13681 939947009 935205856 17248/17161 263 x 1049 x 3407 940123801 930787200 1309/1296 109 x 1801 x 4789 941056273 933918336 4087/4056 157 x 937 x 6397 945959365 630975744 54403/36288 5 x 13 x 19 x 43 x 47 x 379 947993761 767854080 1363/1104 11 x 13 x 37 x 139 x 1289 954732853 930367296 22609/22032 103 x 109 x 277 x 307 955134181 950928720 3861/3844 311 x 1303 x 2357 957044881 843993216 4055/3576 13 x 23 x 1193 x 2683 958735681 934778880 6203/6048 97 x 139 x 211 x 337 958762729 937929696 2163/2116 47 x 3359 x 6073 958970545 696041472 167887/121856 5 x 13 x 113 x 137 x 953 962442001 947635200 65/64 73 x 601 x 21937 962500561 867462400 1509/1360 11 x 239 x 269 x 1361 963163201 691200000 200659/144000 7 x 11 x 13 x 601 x 1601 963168193 877879296 79627/72576 13 x 97 x 757 x 1009 968553181 881798400 256231/233280 31 x 37 x 61 x 109 x 127 968915521 811814400 1289/1080 7 x 61 x 163 x 13921 975303121 836352000 369433/316800 13 x 31 x 41 x 67 x 881 977737321 805049280 787/648 11 x 19 x 37 x 59 x 2143 977892241 864414720 8471/7488 17 x 31 x 53 x 157 x 223 981567505 719913984 2007/1472 5 x 17 x 47 x 277 x 887 981789337 914550912 137737/128304 19 x 73 x 397 x 1783 985052881 972613152 3405/3362 83 x 2953 x 4019 986088961 716636160 42799/31104 11 x 13 x 17 x 19 x 37 x 577 990893569 924844032 92159/86016 17 x 257 x 337 x 673 993420289 952342272 6844/6561 37 x 127 x 163 x 1297 993905641 952560000 43823/42000 41 x 71 x 421 x 811 A small denominator is interesting because the famous "Lehmer's problem" asks if phi(n) ever divides n-1 for a composite number n, i.e., if the denominator is ever 1. It is easy to see that such an n would have to be a Carmichael number. In fact, it is not hard to prove (see below) that if the following conditions are met: a) n is composite b) phi(n) divides 4(n-1) c) n is not an element of {4, 6, 8, 10, 12, 15, 28, 66, 91, 946} then n must be a Carmichael number. Multiples of odd primes are trickier. For example, 4125440/phi(4125441) = 5/3, although 4125441 = 3 x 17 x 23 x 3517 is not a Carmichael number. If k and r are fixed positive integers, r >= 2, then there are only finitely many integers n with exactly r prime factors such that phi(n) divides k(n-1). Here is a proof that if n is a positive integer such that 2(n-1)/phi(n) is an odd integer, then n is 4, 6, or a Carmichael number. The corresponding result with 2 replaced by 4 is not much harder. | Suppose 2(n-1) = k phi(n) where k is odd. This can be written | as 2 = 2n - k phi(n), so gcd(n, phi(n)) is either 1 or 2. | Therefore, n is not divisible by the square of an odd prime. If n | is even, then phi(n) must be == 2 (mod 4), so n is either 4 (OK) | or 2p where p is an odd prime number. It is easy to check that | p = 3 is the only possibility. The remaining case is when n is | odd. Let q be a prime factor of n. Then phi(n) = (q-1) phi(n/q). | Since n can not itself be a prime number, phi(n/q) must be even, | so we have | | n-1 = (q-1) k phi(n/q)/2 | | So n is an odd, squarefree number such that q-1 divides n-1 for | every prime factor q, and therefore n is a Carmichael number. | [] Phil Carmody has found all instances of denominator <= 15 in Pinch's table of Carmichael numbers up to 10^16: 561 320 7/4 3 x 11 x 17 1729 1296 4/3 7 x 13 x 19 2465 1792 11/8 5 x 17 x 29 6601 5280 5/4 7 x 23 x 41 8911 7128 5/4 7 x 19 x 67 15841 12960 11/9 7 x 31 x 73 46657 41472 9/8 13 x 37 x 97 334153 308448 13/12 19 x 43 x 409 449065 326592 11/8 5 x 19 x 29 x 163 2704801 2318400 7/6 11 x 29 x 61 x 139 6840001 5472000 5/4 7 x 17 x 229 x 251 8355841 7372800 17/15 13 x 41 x 61 x 257 19683001 17496000 9/8 13 x 37 x 151 x 271 55462177 49299712 9/8 17 x 23 x 83 x 1709 83099521 74789568 10/9 19 x 37 x 43 x 2749 193708801 152928000 19/15 11 x 13 x 31 x 37 x 1181 206955841 191036160 13/12 17 x 71 x 277 x 619 484662529 430811136 9/8 13 x 29 x 433 x 2969 781347841 732513600 16/15 29 x 61 x 79 x 5591 1752710401 1577439360 10/9 13 x 43 x 631 x 4969 3069196417 2597012352 13/11 7 x 89 x 683 x 7213 4169867689 3706549056 9/8 13 x 29 x 383 x 28879 5064928705 3683584512 11/8 5 x 19 x 37 x 73 x 19739 5985964801 5611842000 16/15 19 x 151 x 271 x 7699 8885251441 7898001280 9/8 11 x 47 x 1109 x 15497 42018333841 37349630080 9/8 11 x 47 x 1049 x 77477 306443088001 286013548800 15/14 19 x 71 x 1361 x 166909 340218584641 309289622400 11/10 13 x 67 x 4591 x 85081 482368435201 452220408000 16/15 19 x 101 x 1949 x 128971 1703236597921 1513988087040 9/8 17 x 23 x 131 x 199 x 167099 2295036426241 2025032140800 17/15 13 x 41 x 107 x 181 x 223 x 997 7485622338601 6653886523200 9/8 11 x 83 x 163 x 239 x 210461 17941025340001 16744956984000 15/14 19 x 127 x 151 x 3221 x 15287 307461980312641 217031986103040 17/12 7 x 11 x 19 x 37 x 59 x 4127 x 23327 436998620298241 407865378945024 15/14 23 x 43 x 1373 x 3697 x 87049 671994263304001 592936114680000 17/15 11 x 59 x 79 x 70981 x 184651 1006604023219201 905943620897280 10/9 11 x 109 x 1597 x 5059 x 103913 1234652134464001 1058273258112000 7/6 11 x 19 x 211 x 47681 x 587179 1584348087168001 1452319079904000 12/11 23 x 31 x 103 x 146609 x 147151 1602991137369601 1335825947808000 6/5 7 x 37 x 1901 x 4091 x 795829 3802910822400001 3422619740160000 10/9 13 x 41 x 1753 x 20149 x 202001 4222760488934401 3941243123005440 15/14 19 x 113 x 167 x 36229 x 325081 6793782344275201 6114404109847680 10/9 11 x 103 x 4297 x 21859 x 63839 7752922144297201 7156543517812800 13/12 19 x 61 x 107 x 17807 x 3510811