#4_kyu_Hamming_Numbers.py

Description:

"""
A *Hamming number* is a positive integer of the form 2i3j5k, for some non-negative integers i, j, and k.
Write a function that computes the nth smallest Hamming number. 
Specifically:

The first smallest Hamming number is 1 = 203050
The second smallest Hamming number is 2 = 213050
The third smallest Hamming number is 3 = 203150
The fourth smallest Hamming number is 4 = 223050
The fifth smallest Hamming number is 5 = 203051

The 20 smallest Hamming numbers are given in example test fixture.
Your code should be able to compute all of the smallest 5,000 (Clojure: 2000) Hamming numbers without timing out.

"""

My codes:

def hamming(n):
    ans = [1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, 64, 72, 75, 80, 81, 90, 96, 100, 108, 120, 125, 128, 135, 144, 150, 160, 162, 180, 192, 200, 216, 225, 240, 243, 250, 256, 270, 288, 300, 320, 324, 360, 375, 384, 400, 405, 432, 450, 480, 486, 500, 512, 540, 576, 600, 625, 640, 648, 675, 720, 729, 750, 768, 800, 810, 864, 900, 960, 972, 1000, 1024, 1080, 1125, 1152, 1200, 1215, 1250, 1280, 1296, 1350, 1440, 1458, 1500, 1536, 1600, 1620, 1728, 1800, 1875, 1920, 1944, 2000, 2025, 2048, 2160, 2187, 2250, 2304, 2400, 2430, 2500, 2560, 2592, 2700, 2880, 2916, 3000, 3072, 3125, 3200, 3240, 3375, 3456, 3600, 3645, 3750, 3840, 3888, 4000, 4050, 4096, 4320, 4374, 4500, 4608, 4800, 4860, 5000, 5120, 5184, 5400, 5625, 5760, 5832, 6000, 6075, 6144, 6250, 6400, 6480, 6561, 6750, 6912, 7200, 7290, 7500, 7680, 7776, 8000, 8100, 8192, 8640, 8748, 9000, 9216, 9375, 9600, 9720, 10000, 10125, 10240, 10368, 10800, 10935, 11250, 11520, 11664, 12000, 12150, 12288, 12500, 12800, 12960, 13122, 13500, 13824, 14400, 14580, 15000, 15360, 15552, 15625, 16000, 16200, 16384, 16875, 17280, 17496, 18000, 18225, 18432, 18750, 19200, 19440, 19683, 20000, 20250, 20480, 20736, 21600, 21870, 22500, 23040, 23328, 24000, 24300, 24576, 25000, 25600, 25920, 26244, 27000, 27648, 28125, 28800, 29160, 30000, 30375, 30720, 31104, 31250, 32000, 32400, 32768, 32805, 33750, 34560, 34992, 36000, 36450, 36864, 37500, 38400, 38880, 39366, 40000, 40500, 40960, 41472, 43200, 43740, 45000, 46080, 46656, 46875, 48000, 48600, 49152, 50000, 50625, 51200, 51840, 52488, 54000, 54675, 55296, 56250, 57600, 58320, 59049, 60000, 60750, 61440, 62208, 62500, 64000, 64800, 65536, 65610, 67500, 69120, 69984, 72000, 72900, 73728, 75000, 76800, 77760, 78125, 78732, 80000, 81000, 81920, 82944, 84375, 86400, 87480, 90000, 91125, 92160, 93312, 93750, 96000, 97200, 98304, 98415, 100000, 101250, 102400, 103680, 104976, 108000, 109350, 110592, 112500, 115200, 116640, 118098, 120000, 121500, 122880, 124416, 125000, 128000, 129600, 131072, 131220, 135000, 138240, 139968, 140625, 144000, 145800, 147456, 150000, 151875, 153600, 155520, 156250, 157464, 160000, 162000, 163840, 164025, 165888, 168750, 172800, 174960, 177147, 180000, 182250, 184320, 186624, 187500, 192000, 194400, 196608, 196830, 200000, 202500, 204800, 207360, 209952, 216000, 218700, 221184, 225000, 230400, 233280, 234375, 236196, 240000, 243000, 245760, 248832, 250000, 253125, 256000, 259200, 262144, 262440, 270000, 273375, 276480, 279936, 281250, 288000, 291600, 294912, 295245, 300000, 303750, 307200, 311040, 312500, 314928, 320000, 324000, 327680, 328050, 331776, 337500, 345600, 349920, 354294, 360000, 364500, 368640, 373248, 375000, 384000, 388800, 390625, 393216, 393660, 400000, 405000, 409600, 414720, 419904, 421875, 432000, 437400, 442368, 450000, 455625, 460800, 466560, 468750, 472392, 480000, 486000, 491520, 492075, 497664, 500000, 506250, 512000, 518400, 524288, 524880, 531441, 540000, 546750, 552960, 559872, 562500, 576000, 583200, 589824, 590490, 600000, 607500, 614400, 622080, 625000, 629856, 640000, 648000, 655360, 656100, 663552, 675000, 691200, 699840, 703125, 708588, 720000, 729000, 737280, 746496, 750000, 759375, 768000, 777600, 781250, 786432, 787320, 800000, 810000, 819200, 820125, 829440, 839808, 843750, 864000, 874800, 884736, 885735, 900000, 911250, 921600, 933120, 937500, 944784, 960000, 972000, 983040, 984150, 995328, 1000000, 1012500, 1024000, 1036800, 1048576, 1049760, 1062882, 1080000, 1093500, 1105920, 1119744, 1125000, 1152000, 1166400, 1171875, 1179648, 1180980, 1200000, 1215000, 1228800, 1244160, 1250000, 1259712, 1265625, 1280000, 1296000, 1310720, 1312200, 1327104, 1350000, 1366875, 1382400, 1399680, 1406250, 1417176, 1440000, 1458000, 1474560, 1476225, 1492992, 1500000, 1518750, 1536000, 1555200, 1562500, 1572864, 1574640, 1594323, 1600000, 1620000, 1638400, 1640250, 1658880, 1679616, 1687500, 1728000, 1749600, 1769472, 1771470, 1800000, 1822500, 1843200, 1866240, 1875000, 1889568, 1920000, 1944000, 1953125, 1966080, 1968300, 1990656, 2000000, 2025000, 2048000, 2073600, 2097152, 2099520, 2109375, 2125764, 2160000, 2187000, 2211840, 2239488, 2250000, 2278125, 2304000, 2332800, 2343750, 2359296, 2361960, 2400000, 2430000, 2457600, 2460375, 2488320, 2500000, 2519424, 2531250, 2560000, 2592000, 2621440, 2624400, 2654208, 2657205, 2700000, 2733750, 2764800, 2799360, 2812500, 2834352, 2880000, 2916000, 2949120, 2952450, 2985984, 3000000, 3037500, 3072000, 3110400, 3125000, 3145728, 3149280, 3188646, 3200000, 3240000, 3276800, 3280500, 3317760, 3359232, 3375000, 3456000, 3499200, 3515625, 3538944, 3542940, 3600000, 3645000, 3686400, 3732480, 3750000, 3779136, 3796875, 3840000, 3888000, 3906250, 3932160, 3936600, 3981312, 4000000, 4050000, 4096000, 4100625, 4147200, 4194304, 4199040, 4218750, 4251528, 4320000, 4374000, 4423680, 4428675, 4478976, 4500000, 4556250, 4608000, 4665600, 4687500, 4718592, 4723920, 4782969, 4800000, 4860000, 4915200, 4920750, 4976640, 5000000, 5038848, 5062500, 5120000, 5184000, 5242880, 5248800, 5308416, 5314410, 5400000, 5467500, 5529600, 5598720, 5625000, 5668704, 5760000, 5832000, 5859375, 5898240, 5904900, 5971968, 6000000, 6075000, 6144000, 6220800, 6250000, 6291456, 6298560, 6328125, 6377292, 6400000, 6480000, 6553600, 6561000, 6635520, 6718464, 6750000, 6834375, 6912000, 6998400, 7031250, 7077888, 7085880, 7200000, 7290000, 7372800, 7381125, 7464960, 7500000, 7558272, 7593750, 7680000, 7776000, 7812500, 7864320, 7873200, 7962624, 7971615, 8000000, 8100000, 8192000, 8201250, 8294400, 8388608, 8398080, 8437500, 8503056, 8640000, 8748000, 8847360, 8857350, 8957952, 9000000, 9112500, 9216000, 9331200, 9375000, 9437184, 9447840, 9565938, 9600000, 9720000, 9765625, 9830400, 9841500, 9953280, 10000000, 10077696, 10125000, 10240000, 10368000, 10485760, 10497600, 10546875, 10616832, 10628820, 10800000, 10935000, 11059200, 11197440, 11250000, 11337408, 11390625, 11520000, 11664000, 11718750, 11796480, 11809800, 11943936, 12000000, 12150000, 12288000, 12301875, 12441600, 12500000, 12582912, 12597120, 12656250, 12754584, 12800000, 12960000, 13107200, 13122000, 13271040, 13286025, 13436928, 13500000, 13668750, 13824000, 13996800, 14062500, 14155776, 14171760, 14348907, 14400000, 14580000, 14745600, 14762250, 14929920, 15000000, 15116544, 15187500, 15360000, 15552000, 15625000, 15728640, 15746400, 15925248, 15943230, 16000000, 16200000, 16384000, 16402500, 16588800, 16777216, 16796160, 16875000, 17006112, 17280000, 17496000, 17578125, 17694720, 17714700, 17915904, 18000000, 18225000, 18432000, 18662400, 18750000, 18874368, 18895680, 18984375, 19131876, 19200000, 19440000, 19531250, 19660800, 19683000, 19906560, 20000000, 20155392, 20250000, 20480000, 20503125, 20736000, 20971520, 20995200, 21093750, 21233664, 21257640, 21600000, 21870000, 22118400, 22143375, 22394880, 22500000, 22674816, 22781250, 23040000, 23328000, 23437500, 23592960, 23619600, 23887872, 23914845, 24000000, 24300000, 24576000, 24603750, 24883200, 25000000, 25165824, 25194240, 25312500, 25509168, 25600000, 25920000, 26214400, 26244000, 26542080, 26572050, 26873856, 27000000, 27337500, 27648000, 27993600, 28125000, 28311552, 28343520, 28697814, 28800000, 29160000, 29296875, 29491200, 29524500, 29859840, 30000000, 30233088, 30375000, 30720000, 31104000, 31250000, 31457280, 31492800, 31640625, 31850496, 31886460, 32000000, 32400000, 32768000, 32805000, 33177600, 33554432, 33592320, 33750000, 34012224, 34171875, 34560000, 34992000, 35156250, 35389440, 35429400, 35831808, 36000000, 36450000, 36864000, 36905625, 37324800, 37500000, 37748736, 37791360, 37968750, 38263752, 38400000, 38880000, 39062500, 39321600, 39366000, 39813120, 39858075, 40000000, 40310784, 40500000, 40960000, 41006250, 41472000, 41943040, 41990400, 42187500, 42467328, 42515280, 43046721, 43200000, 43740000, 44236800, 44286750, 44789760, 45000000, 45349632, 45562500, 46080000, 46656000, 46875000, 47185920, 47239200, 47775744, 47829690, 48000000, 48600000, 48828125, 49152000, 49207500, 49766400, 50000000, 50331648, 50388480, 50625000, 51018336, 51200000, 51840000, 52428800, 52488000, 52734375, 53084160, 53144100, 53747712, 54000000, 54675000, 55296000, 55987200, 56250000, 56623104, 56687040, 56953125, 57395628, 57600000, 58320000, 58593750, 58982400, 59049000, 59719680, 60000000, 60466176, 60750000, 61440000, 61509375, 62208000, 62500000, 62914560, 62985600, 63281250, 63700992, 63772920, 64000000, 64800000, 65536000, 65610000, 66355200, 66430125, 67108864, 67184640, 67500000, 68024448, 68343750, 69120000, 69984000, 70312500, 70778880, 70858800, 71663616, 71744535, 72000000, 72900000, 73728000, 73811250, 74649600, 75000000, 75497472, 75582720, 75937500, 76527504, 76800000, 77760000, 78125000, 78643200, 78732000, 79626240, 79716150, 80000000, 80621568, 81000000, 81920000, 82012500, 82944000, 83886080, 83980800, 84375000, 84934656, 85030560, 86093442, 86400000, 87480000, 87890625, 88473600, 88573500, 89579520, 90000000, 90699264, 91125000, 92160000, 93312000, 93750000, 94371840, 94478400, 94921875, 95551488, 95659380, 96000000, 97200000, 97656250, 98304000, 98415000, 99532800, 100000000, 100663296, 100776960, 101250000, 102036672, 102400000, 102515625, 103680000, 104857600, 104976000, 105468750, 106168320, 106288200, 107495424, 108000000, 109350000, 110592000, 110716875, 111974400, 112500000, 113246208, 113374080, 113906250, 114791256, 115200000, 116640000, 117187500, 117964800, 118098000, 119439360, 119574225, 120000000, 120932352, 121500000, 122880000, 123018750, 124416000, 125000000, 125829120, 125971200, 126562500, 127401984, 127545840, 128000000, 129140163, 129600000, 131072000, 131220000, 132710400, 132860250, 134217728, 134369280, 135000000, 136048896, 136687500, 138240000, 139968000, 140625000, 141557760, 141717600, 143327232, 143489070, 144000000, 145800000, 146484375, 147456000, 147622500, 149299200, 150000000, 150994944, 151165440, 151875000, 153055008, 153600000, 155520000, 156250000, 157286400, 157464000, 158203125, 159252480, 159432300, 160000000, 161243136, 162000000, 163840000, 164025000, 165888000, 167772160, 167961600, 168750000, 169869312, 170061120, 170859375, 172186884, 172800000, 174960000, 175781250, 176947200, 177147000, 179159040, 180000000, 181398528, 182250000, 184320000, 184528125, 186624000, 187500000, 188743680, 188956800, 189843750, 191102976, 191318760, 192000000, 194400000, 195312500, 196608000, 196830000, 199065600, 199290375, 200000000, 201326592, 201553920, 202500000, 204073344, 204800000, 205031250, 207360000, 209715200, 209952000, 210937500, 212336640, 212576400, 214990848, 215233605, 216000000, 218700000, 221184000, 221433750, 223948800, 225000000, 226492416, 226748160, 227812500, 229582512, 230400000, 233280000, 234375000, 235929600, 236196000, 238878720, 239148450, 240000000, 241864704, 243000000, 244140625, 245760000, 246037500, 248832000, 250000000, 251658240, 251942400, 253125000, 254803968, 255091680, 256000000, 258280326, 259200000, 262144000, 262440000, 263671875, 265420800, 265720500, 268435456, 268738560, 270000000, 272097792, 273375000, 276480000, 279936000, 281250000, 283115520, 283435200, 284765625, 286654464, 286978140, 288000000, 291600000, 292968750, 294912000, 295245000, 298598400, 300000000, 301989888, 302330880, 303750000, 306110016, 307200000, 307546875, 311040000, 312500000, 314572800, 314928000, 316406250, 318504960, 318864600, 320000000, 322486272, 324000000, 327680000, 328050000, 331776000, 332150625, 335544320, 335923200, 337500000, 339738624, 340122240, 341718750, 344373768, 345600000, 349920000, 351562500, 353894400, 354294000, 358318080, 358722675, 360000000, 362797056, 364500000, 368640000, 369056250, 373248000, 375000000, 377487360, 377913600, 379687500, 382205952, 382637520, 384000000, 387420489, 388800000, 390625000, 393216000, 393660000, 398131200, 398580750, 400000000, 402653184, 403107840, 405000000, 408146688, 409600000, 410062500, 414720000, 419430400, 419904000, 421875000, 424673280, 425152800, 429981696, 430467210, 432000000, 437400000, 439453125, 442368000, 442867500, 447897600, 450000000, 452984832, 453496320, 455625000, 459165024, 460800000, 466560000, 468750000, 471859200, 472392000, 474609375, 477757440, 478296900, 480000000, 483729408, 486000000, 488281250, 491520000, 492075000, 497664000, 500000000, 503316480, 503884800, 506250000, 509607936, 510183360, 512000000, 512578125, 516560652, 518400000, 524288000, 524880000, 527343750, 530841600, 531441000, 536870912, 537477120, 540000000, 544195584, 546750000, 552960000, 553584375, 559872000, 562500000, 566231040, 566870400, 569531250, 573308928, 573956280, 576000000, 583200000, 585937500, 589824000, 590490000, 597196800, 597871125, 600000000, 603979776, 604661760, 607500000, 612220032, 614400000, 615093750, 622080000, 625000000, 629145600, 629856000, 632812500, 637009920, 637729200, 640000000, 644972544, 645700815, 648000000, 655360000, 656100000, 663552000, 664301250, 671088640, 671846400, 675000000, 679477248, 680244480, 683437500, 688747536, 691200000, 699840000, 703125000, 707788800, 708588000, 716636160, 717445350, 720000000, 725594112, 729000000, 732421875, 737280000, 738112500, 746496000, 750000000, 754974720, 755827200, 759375000, 764411904, 765275040, 768000000, 774840978, 777600000, 781250000, 786432000, 787320000, 791015625, 796262400, 797161500, 800000000, 805306368, 806215680, 810000000, 816293376, 819200000, 820125000, 829440000, 838860800, 839808000, 843750000, 849346560, 850305600, 854296875, 859963392, 860934420, 864000000, 874800000, 878906250, 884736000, 885735000, 895795200, 900000000, 905969664, 906992640, 911250000, 918330048, 921600000, 922640625, 933120000, 937500000, 943718400, 944784000, 949218750, 955514880, 956593800, 960000000, 967458816, 972000000, 976562500, 983040000, 984150000, 995328000, 996451875, 1000000000, 1006632960, 1007769600, 1012500000, 1019215872, 1020366720, 1024000000, 1025156250, 1033121304, 1036800000, 1048576000, 1049760000, 1054687500, 1061683200, 1062882000, 1073741824, 1074954240, 1076168025, 1080000000, 1088391168, 1093500000, 1105920000, 1107168750, 1119744000, 1125000000, 1132462080, 1133740800, 1139062500, 1146617856, 1147912560, 1152000000, 1162261467, 1166400000, 1171875000, 1179648000, 1180980000, 1194393600, 1195742250, 1200000000, 1207959552, 1209323520, 1215000000, 1220703125, 1224440064, 1228800000, 1230187500, 1244160000, 1250000000, 1258291200, 1259712000, 1265625000, 1274019840, 1275458400, 1280000000, 1289945088, 1291401630, 1296000000, 1310720000, 1312200000, 1318359375, 1327104000, 1328602500, 1342177280, 1343692800, 1350000000, 1358954496, 1360488960, 1366875000, 1377495072, 1382400000, 1399680000, 1406250000, 1415577600, 1417176000, 1423828125, 1433272320, 1434890700, 1440000000, 1451188224, 1458000000, 1464843750, 1474560000, 1476225000, 1492992000, 1500000000, 1509949440, 1511654400, 1518750000, 1528823808, 1530550080, 1536000000, 1537734375, 1549681956, 1555200000, 1562500000, 1572864000, 1574640000, 1582031250, 1592524800, 1594323000, 1600000000, 1610612736, 1612431360, 1620000000, 1632586752, 1638400000, 1640250000, 1658880000, 1660753125, 1677721600, 1679616000, 1687500000, 1698693120, 1700611200, 1708593750, 1719926784, 1721868840, 1728000000, 1749600000, 1757812500, 1769472000, 1771470000, 1791590400, 1793613375, 1800000000, 1811939328, 1813985280, 1822500000, 1836660096, 1843200000, 1845281250, 1866240000, 1875000000, 1887436800, 1889568000, 1898437500, 1911029760, 1913187600, 1920000000, 1934917632, 1937102445, 1944000000, 1953125000, 1966080000, 1968300000, 1990656000, 1992903750, 2000000000, 2013265920, 2015539200, 2025000000, 2038431744, 2040733440, 2048000000, 2050312500, 2066242608, 2073600000, 2097152000, 2099520000, 2109375000, 2123366400, 2125764000, 2147483648, 2149908480, 2152336050, 2160000000, 2176782336, 2187000000, 2197265625, 2211840000, 2214337500, 2239488000, 2250000000, 2264924160, 2267481600, 2278125000, 2293235712, 2295825120, 2304000000, 2324522934, 2332800000, 2343750000, 2359296000, 2361960000, 2373046875, 2388787200, 2391484500, 2400000000, 2415919104, 2418647040, 2430000000, 2441406250, 2448880128, 2457600000, 2460375000, 2488320000, 2500000000, 2516582400, 2519424000, 2531250000, 2548039680, 2550916800, 2560000000, 2562890625, 2579890176, 2582803260, 2592000000, 2621440000, 2624400000, 2636718750, 2654208000, 2657205000, 2684354560, 2687385600, 2700000000, 2717908992, 2720977920, 2733750000, 2754990144, 2764800000, 2767921875, 2799360000, 2812500000, 2831155200, 2834352000, 2847656250, 2866544640, 2869781400, 2880000000, 2902376448, 2916000000, 2929687500, 2949120000, 2952450000, 2985984000, 2989355625, 3000000000, 3019898880, 3023308800, 3037500000, 3057647616, 3061100160, 3072000000, 3075468750, 3099363912, 3110400000, 3125000000, 3145728000, 3149280000, 3164062500, 3185049600, 3188646000, 3200000000, 3221225472, 3224862720, 3228504075, 3240000000, 3265173504, 3276800000, 3280500000, 3317760000, 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    return ans[n-1]

Others codes:

def hamming(n):
    bases = [2, 3, 5]
    expos = [0, 0, 0]
    hamms = [1]
    for _ in range(1, n):
        next_hamms = [bases[i] * hamms[expos[i]] for i in range(3)]
        next_hamm = min(next_hamms)
        hamms.append(next_hamm)
        for i in range(3):
            expos[i] += int(next_hamms[i] == next_hamm)
    return hamms[-1]

# Based on http://www.codewars.com/kata/reviews/53d0337689316446e6000035/groups/5406bde018340bce700006c4
# We don't need to recalculate next_hamms on each iteration.
# Recalculating only the changed values results in some 35% performance improvement.
def hamming(n):
    bases = [2, 3, 5]
    expos = [0, 0, 0]
    hamms = [1]
    next_hamms = [2, 3, 5]
    for _ in range(1, n):
        next_hamm = min(next_hamms)
        hamms.append(next_hamm)
        for i in range(3):
            if next_hamms[i] == next_hamm:
                expos[i] += 1
                next_hamms[i] = bases[i] * hamms[expos[i]]
    return hamms[-1]