Asymptotic factorizations of an Eulerproduct into primefactors.

Let   : :   where the q are prime. We are considering limN->oo wN,s (mod pk) where the pk are primes and here pk=2...89

and estimate the relative frequency of primefactors pk after evaluating the euler-product wN,s with exponent s up to the Nth prime.

Tables: absolute and (scaled) relative count of occurences of primefactor pk in relation to N, formula: rel=(count/N) * (pk-1)²

 N=200 Count of occurences of primefactor p in the (partial) Eulerproduct over prime(1)...prime(200) s\p 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 1 392 142 61 35 20 17 12 9 10 6 4 3 3 4 5 3 4 3 1 2 3 1 4 2 2 784 294 114 71 40 35 23 20 17 12 11 8 7 10 9 6 6 6 3 4 4 4 6 2 3 392 238 61 107 20 50 12 35 10 6 18 14 3 12 5 3 4 7 8 2 10 5 4 2 4 983 294 245 71 40 75 48 20 17 27 11 18 16 10 9 13 6 13 3 4 8 4 6 6 5 392 142 110 35 107 17 12 9 10 6 32 3 21 4 5 3 4 14 1 14 3 1 4 2 6 784 493 114 224 40 104 23 69 17 12 40 28 7 28 9 6 6 17 15 4 18 13 6 2 7 392 142 61 67 20 17 12 9 10 54 4 3 3 31 5 3 4 3 1 18 3 1 4 2 8 1182 294 245 71 40 75 99 20 17 27 11 18 38 10 9 13 6 13 3 4 16 4 6 15 9 392 334 61 107 20 50 12 105 10 6 18 52 3 12 5 3 4 7 8 2 24 5 4 2 10 784 294 208 71 217 35 23 20 17 12 69 8 50 10 9 6 6 33 3 28 4 4 6 2 11 392 142 61 35 39 17 12 9 98 6 4 3 3 4 5 3 4 3 36 2 3 1 4 24 12 983 493 245 224 40 216 48 69 17 27 40 61 16 28 9 13 6 37 15 4 34 13 6 6 relative frequency: count/N*(pk - 1)² 1 1.96 2.84 4.88 6.30 10.00 12.24 15.36 14.58 24.20 23.52 18.00 19.44 24.00 35.28 52.90 40.56 67.28 54.00 21.78 49.00 77.76 30.42 134.48 77.44 2 3.92 5.88 9.12 12.78 20.00 25.20 29.44 32.40 41.14 47.04 49.50 51.84 56.00 88.20 95.22 81.12 100.92 108.00 65.34 98.00 103.68 121.68 201.72 77.44 3 1.96 4.76 4.88 19.26 10.00 36.00 15.36 56.70 24.20 23.52 81.00 90.72 24.00 105.84 52.90 40.56 67.28 126.00 174.24 49.00 259.20 152.10 134.48 77.44 4 4.92 5.88 19.60 12.78 20.00 54.00 61.44 32.40 41.14 105.84 49.50 116.64 128.00 88.20 95.22 175.76 100.92 234.00 65.34 98.00 207.36 121.68 201.72 232.32 5 1.96 2.84 8.80 6.30 53.50 12.24 15.36 14.58 24.20 23.52 144.00 19.44 168.00 35.28 52.90 40.56 67.28 252.00 21.78 343.00 77.76 30.42 134.48 77.44 6 3.92 9.86 9.12 40.32 20.00 74.88 29.44 111.78 41.14 47.04 180.00 181.44 56.00 246.96 95.22 81.12 100.92 306.00 326.70 98.00 466.56 395.46 201.72 77.44 7 1.96 2.84 4.88 12.06 10.00 12.24 15.36 14.58 24.20 211.68 18.00 19.44 24.00 273.42 52.90 40.56 67.28 54.00 21.78 441.00 77.76 30.42 134.48 77.44 8 5.91 5.88 19.60 12.78 20.00 54.00 126.72 32.40 41.14 105.84 49.50 116.64 304.00 88.20 95.22 175.76 100.92 234.00 65.34 98.00 414.72 121.68 201.72 580.80 9 1.96 6.68 4.88 19.26 10.00 36.00 15.36 170.10 24.20 23.52 81.00 336.96 24.00 105.84 52.90 40.56 67.28 126.00 174.24 49.00 622.08 152.10 134.48 77.44 10 3.92 5.88 16.64 12.78 108.50 25.20 29.44 32.40 41.14 47.04 310.50 51.84 400.00 88.20 95.22 81.12 100.92 594.00 65.34 686.00 103.68 121.68 201.72 77.44 11 1.96 2.84 4.88 6.30 19.50 12.24 15.36 14.58 237.16 23.52 18.00 19.44 24.00 35.28 52.90 40.56 67.28 54.00 784.08 49.00 77.76 30.42 134.48 929.28 12 4.92 9.86 19.60 40.32 20.00 155.52 61.44 111.78 41.14 105.84 180.00 395.28 128.00 246.96 95.22 175.76 100.92 666.00 326.70 98.00 881.28 395.46 201.72 232.32 113

 N=1000 Count of occurences of primefactor p in the (partial) Eulerproduct over prime(1)...prime(1000) s\p 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 1 1961 736 307 191 110 85 65 50 48 33 31 28 24 24 23 17 19 17 18 13 14 13 11 12 2 3952 1494 613 385 219 173 129 110 96 68 67 54 48 48 44 36 35 30 34 27 24 25 21 21 3 1961 1226 307 573 110 261 65 175 48 33 103 88 24 73 23 17 19 54 50 13 42 40 11 12 4 4951 1494 1253 385 219 356 265 110 96 144 67 107 97 48 44 74 35 65 34 27 49 25 21 45 5 1961 736 552 191 545 85 65 50 48 33 176 28 124 24 23 17 19 83 18 69 14 13 11 12 6 3952 2493 613 1160 219 527 129 349 96 68 205 165 48 142 44 36 35 98 95 27 79 78 21 21 7 1961 736 307 356 110 85 65 50 48 258 31 28 24 166 23 17 19 17 18 98 14 13 11 12 8 5950 1494 1253 385 219 356 523 110 96 144 67 107 200 48 44 74 35 65 34 27 103 25 21 81 9 1961 1716 307 573 110 261 65 513 48 33 103 262 24 73 23 17 19 54 50 13 119 40 11 12 10 3952 1494 1104 385 1090 173 129 110 96 68 347 54 253 48 44 36 35 166 34 138 24 25 21 21 11 1961 736 307 191 211 85 65 50 510 33 31 28 24 24 23 17 19 17 176 13 14 13 11 129 12 4951 2493 1253 1160 219 1080 265 349 96 144 205 334 97 142 44 74 35 202 95 27 163 78 21 45 relative frequency: count/N*(pk - 1)² 1 1.96 2.94 4.91 6.88 11.00 12.24 16.64 16.20 23.23 25.87 27.90 36.29 38.40 42.34 48.67 45.97 63.92 61.20 78.41 63.70 72.58 79.09 73.96 92.93 2 3.95 5.98 9.81 13.86 21.90 24.91 33.02 35.64 46.46 53.31 60.30 69.98 76.80 84.67 93.10 97.34 117.74 108.00 148.10 132.30 124.42 152.10 141.20 162.62 3 1.96 4.90 4.91 20.63 11.00 37.58 16.64 56.70 23.23 25.87 92.70 114.05 38.40 128.77 48.67 45.97 63.92 194.40 217.80 63.70 217.73 243.36 73.96 92.93 4 4.95 5.98 20.05 13.86 21.90 51.26 67.84 35.64 46.46 112.90 60.30 138.67 155.20 84.67 93.10 200.10 117.74 234.00 148.10 132.30 254.02 152.10 141.20 348.48 5 1.96 2.94 8.83 6.88 54.50 12.24 16.64 16.20 23.23 25.87 158.40 36.29 198.40 42.34 48.67 45.97 63.92 298.80 78.41 338.10 72.58 79.09 73.96 92.93 6 3.95 9.97 9.81 41.76 21.90 75.89 33.02 113.08 46.46 53.31 184.50 213.84 76.80 250.49 93.10 97.34 117.74 352.80 413.82 132.30 409.54 474.55 141.20 162.62 7 1.96 2.94 4.91 12.82 11.00 12.24 16.64 16.20 23.23 202.27 27.90 36.29 38.40 292.82 48.67 45.97 63.92 61.20 78.41 480.20 72.58 79.09 73.96 92.93 8 5.95 5.98 20.05 13.86 21.90 51.26 133.89 35.64 46.46 112.90 60.30 138.67 320.00 84.67 93.10 200.10 117.74 234.00 148.10 132.30 533.95 152.10 141.20 627.26 9 1.96 6.86 4.91 20.63 11.00 37.58 16.64 166.21 23.23 25.87 92.70 339.55 38.40 128.77 48.67 45.97 63.92 194.40 217.80 63.70 616.90 243.36 73.96 92.93 10 3.95 5.98 17.66 13.86 109.00 24.91 33.02 35.64 46.46 53.31 312.30 69.98 404.80 84.67 93.10 97.34 117.74 597.60 148.10 676.20 124.42 152.10 141.20 162.62 11 1.96 2.94 4.91 6.88 21.10 12.24 16.64 16.20 246.84 25.87 27.90 36.29 38.40 42.34 48.67 45.97 63.92 61.20 766.66 63.70 72.58 79.09 73.96 998.98 12 4.95 9.97 20.05 41.76 21.90 155.52 67.84 113.08 46.46 112.90 184.50 432.86 155.20 250.49 93.10 200.10 117.74 727.20 413.82 132.30 844.99 474.55 141.20 348.48

 N=2000 Count of occurences of primefactor p in the (partial) Eulerproduct over prime(1)...prime(2000) s\p 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 1 3965 1479 617 384 221 183 132 107 91 75 62 55 50 47 46 34 36 35 31 29 28 27 26 21 2 7979 2986 1226 777 440 359 259 233 184 146 129 112 97 97 86 73 67 65 55 60 53 51 47 43 3 3965 2469 617 1154 221 541 132 347 91 75 205 170 50 152 46 34 36 106 90 29 85 82 26 21 4 9978 2986 2494 777 440 721 522 233 184 296 129 230 199 97 86 152 67 134 55 60 105 51 47 90 5 3965 1479 1112 384 1102 183 132 107 91 75 332 55 244 47 46 34 36 162 31 143 28 27 26 21 6 7979 4985 1226 2333 440 1072 259 711 184 146 406 340 97 298 86 73 67 201 174 60 168 165 47 43 7 3965 1479 617 716 221 183 132 107 91 505 62 55 50 338 46 34 36 35 31 199 28 27 26 21 8 11977 2986 2494 777 440 721 1059 233 184 296 129 230 403 97 86 152 67 134 55 60 212 51 47 179 9 3965 3459 617 1154 221 541 132 1034 91 75 205 503 50 152 46 34 36 106 90 29 252 82 26 21 10 7979 2986 2210 777 2195 359 259 233 184 146 673 112 504 97 86 73 67 330 55 284 53 51 47 43 11 3965 1479 617 384 420 183 132 107 1040 75 62 55 50 47 46 34 36 35 340 29 28 27 26 250 12 9978 4985 2494 2333 440 2166 522 711 184 296 406 685 199 298 86 152 67 408 174 60 334 165 47 90 relative frequency: count/N*(pk - 1)² 1 1.98 2.96 4.94 6.91 11.05 13.18 16.90 17.33 22.02 29.40 27.90 35.64 40.00 41.45 48.67 45.97 60.55 63.00 67.52 71.05 72.58 82.13 87.41 81.31 2 3.99 5.97 9.81 13.99 22.00 25.85 33.15 37.75 44.53 57.23 58.05 72.58 77.60 85.55 90.99 98.70 112.69 117.00 119.79 147.00 137.38 155.14 158.01 166.50 3 1.98 4.94 4.94 20.77 11.05 38.95 16.90 56.21 22.02 29.40 92.25 110.16 40.00 134.06 48.67 45.97 60.55 190.80 196.02 71.05 220.32 249.44 87.41 81.31 4 4.99 5.97 19.95 13.99 22.00 51.91 66.82 37.75 44.53 116.03 58.05 149.04 159.20 85.55 90.99 205.50 112.69 241.20 119.79 147.00 272.16 155.14 158.01 348.48 5 1.98 2.96 8.90 6.91 55.10 13.18 16.90 17.33 22.02 29.40 149.40 35.64 195.20 41.45 48.67 45.97 60.55 291.60 67.52 350.35 72.58 82.13 87.41 81.31 6 3.99 9.97 9.81 41.99 22.00 77.18 33.15 115.18 44.53 57.23 182.70 220.32 77.60 262.84 90.99 98.70 112.69 361.80 378.97 147.00 435.46 501.93 158.01 166.50 7 1.98 2.96 4.94 12.89 11.05 13.18 16.90 17.33 22.02 197.96 27.90 35.64 40.00 298.12 48.67 45.97 60.55 63.00 67.52 487.55 72.58 82.13 87.41 81.31 8 5.99 5.97 19.95 13.99 22.00 51.91 135.55 37.75 44.53 116.03 58.05 149.04 322.40 85.55 90.99 205.50 112.69 241.20 119.79 147.00 549.50 155.14 158.01 693.09 9 1.98 6.92 4.94 20.77 11.05 38.95 16.90 167.51 22.02 29.40 92.25 325.94 40.00 134.06 48.67 45.97 60.55 190.80 196.02 71.05 653.18 249.44 87.41 81.31 10 3.99 5.97 17.68 13.99 109.75 25.85 33.15 37.75 44.53 57.23 302.85 72.58 403.20 85.55 90.99 98.70 112.69 594.00 119.79 695.80 137.38 155.14 158.01 166.50 11 1.98 2.96 4.94 6.91 21.00 13.18 16.90 17.33 251.68 29.40 27.90 35.64 40.00 41.45 48.67 45.97 60.55 63.00 740.52 71.05 72.58 82.13 87.41 968.00 12 4.99 9.97 19.95 41.99 22.00 155.95 66.82 115.18 44.53 116.03 182.70 443.88 159.20 262.84 90.99 205.50 112.69 734.40 378.97 147.00 865.73 501.93 158.01 348.48

 N=10000 Count of occurences of primefactor p in the (partial) Eulerproduct over prime(1)...prime(10000) s\p 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 1 19954 7498 3110 1922 1086 898 660 574 464 369 348 280 257 251 219 193 179 172 150 153 138 134 115 106 2 39960 14984 6211 3857 2202 1789 1322 1169 941 738 688 567 515 489 444 385 347 346 294 293 279 257 238 221 3 19954 12486 3110 5783 1086 2702 660 1750 464 369 1047 863 257 739 219 193 179 516 458 153 423 390 115 106 4 49959 14984 12495 3857 2202 3586 2640 1169 941 1489 688 1137 1024 489 444 784 347 678 294 293 561 257 238 442 5 19954 7498 5594 1922 5487 898 660 574 464 369 1737 280 1257 251 219 193 179 835 150 737 138 134 115 106 6 39960 24983 6211 11642 2202 5408 1322 3520 941 738 2080 1714 515 1443 444 385 347 1030 915 293 848 774 238 221 7 19954 7498 3110 3583 1086 898 660 574 464 2566 348 280 257 1712 219 193 179 172 150 1006 138 134 115 106 8 59958 14984 12495 3857 2202 3586 5297 1169 941 1489 688 1137 2045 489 444 784 347 678 294 293 1130 257 238 899 9 19954 17474 3110 5783 1086 2702 660 5274 464 369 1047 2554 257 739 219 193 179 516 458 153 1268 390 115 106 10 39960 14984 11186 3857 11007 1789 1322 1169 941 738 3443 567 2561 489 444 385 347 1689 294 1442 279 257 238 221 11 19954 7498 3110 1922 2076 898 660 574 5204 369 348 280 257 251 219 193 179 172 1680 153 138 134 115 1245 12 49959 24983 12495 11642 2202 10840 2640 3520 941 1489 2080 3434 1024 1443 444 784 347 2041 915 293 1700 774 238 442 relative frequency rounded to integer: count/N*(pk - 1)² 1 2 3 5 7 11 13 17 19 22 29 31 36 41 44 46 52 60 62 65 75 72 82 77 82 2 4 6 10 14 22 26 34 38 46 58 62 73 82 86 94 104 117 125 128 144 145 156 160 171 3 2 5 5 21 11 39 17 57 22 29 94 112 41 130 46 52 60 186 200 75 219 237 77 82 4 5 6 20 14 22 52 68 38 46 117 62 147 164 86 94 212 117 244 128 144 291 156 160 342 5 2 3 9 7 55 13 17 19 22 29 156 36 201 44 46 52 60 301 65 361 72 82 77 82 6 4 10 10 42 22 78 34 114 46 58 187 222 82 255 94 104 117 371 399 144 440 471 160 171 7 2 3 5 13 11 13 17 19 22 201 31 36 41 302 46 52 60 62 65 493 72 82 77 82 8 6 6 20 14 22 52 136 38 46 117 62 147 327 86 94 212 117 244 128 144 586 156 160 696 9 2 7 5 21 11 39 17 171 22 29 94 331 41 130 46 52 60 186 200 75 657 237 77 82 10 4 6 18 14 110 26 34 38 46 58 310 73 410 86 94 104 117 608 128 707 145 156 160 171 11 2 3 5 7 21 13 17 19 252 29 31 36 41 44 46 52 60 62 732 75 72 82 77 964 12 5 10 20 42 22 156 68 114 46 117 187 445 164 255 94 212 117 735 399 144 881 471 160 342

 N=20000 Count of occurences of primefactor p in the (partial) Eulerproduct over prime(1)...prime(20000) s\p 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 1 40152 15083 6247 3897 2195 1812 1332 1170 946 742 683 574 524 489 444 394 357 336 307 294 282 273 236 231 2 80415 30152 12538 7804 4393 3613 2665 2351 1902 1481 1380 1147 1042 984 898 786 711 691 614 578 555 528 491 475 3 40152 25127 6247 11690 2195 5402 1332 3552 946 742 2063 1724 524 1511 444 394 357 1012 917 294 841 779 236 231 4 100537 30152 25157 7804 4393 7249 5351 2351 1902 2978 1380 2289 2074 984 898 1581 711 1376 614 578 1132 528 491 929 5 40152 15083 11256 3897 11040 1812 1332 1170 946 742 3454 574 2544 489 444 394 357 1707 307 1459 282 273 236 231 6 80415 50274 12538 23461 4393 10852 2665 7069 1902 1481 4169 3440 1042 2965 898 786 711 2043 1856 578 1671 1558 491 475 7 40152 15083 6247 7245 2195 1812 1332 1170 946 5188 683 574 524 3419 444 394 357 336 307 2005 282 273 236 231 8 120659 30152 25157 7804 4393 7249 10673 2351 1902 2978 1380 2289 4142 984 898 1581 711 1376 614 578 2257 528 491 1835 9 40152 35171 6247 11690 2195 5402 1332 10600 946 742 2063 5128 524 1511 444 394 357 1012 917 294 2574 779 236 231 10 80415 30152 22571 7804 22113 3613 2665 2351 1902 1481 6912 1147 5150 984 898 786 711 3423 614 2903 555 528 491 475 11 40152 15083 6247 3897 4201 1812 1332 1170 10477 742 683 574 524 489 444 394 357 336 3378 294 282 273 236 2541 12 100537 50274 25157 23461 4393 21787 5351 7069 1902 2978 4169 6928 2074 2965 898 1581 711 4077 1856 578 3374 1558 491 929 relative frequency rounded to integer: count/N*(pk - 1)² 1 2 3 5 7 11 13 17 19 23 29 31 37 42 43 47 53 60 60 67 72 73 83 79 89 2 4 6 10 14 22 26 34 38 46 58 62 74 83 87 95 106 120 124 134 142 144 161 165 184 3 2 5 5 21 11 39 17 58 23 29 93 112 42 133 47 53 60 182 200 72 218 237 79 89 p=1(mod3) 4 5 6 20 14 22 52 68 38 46 117 62 148 166 87 95 214 120 248 134 142 293 161 165 360 5 2 3 9 7 55 13 17 19 23 29 155 37 204 43 47 53 60 307 67 357 73 83 79 89 p=1(mod5) 6 4 10 10 42 22 78 34 115 46 58 188 223 83 262 95 106 120 368 404 142 433 474 165 184 7 2 3 5 13 11 13 17 19 23 203 31 37 42 302 47 53 60 60 67 491 73 83 79 89 p=1(mod7) 8 6 6 20 14 22 52 137 38 46 117 62 148 331 87 95 214 120 248 134 142 585 161 165 711 9 2 7 5 21 11 39 17 172 23 29 93 332 42 133 47 53 60 182 200 72 667 237 79 89 10 4 6 18 14 111 26 34 38 46 58 311 74 412 87 95 106 120 616 134 711 144 161 165 184 11 2 3 5 7 21 13 17 19 254 29 31 37 42 43 47 53 60 60 736 72 73 83 79 984 12 5 10 20 42 22 157 68 115 46 117 188 449 166 262 95 214 120 734 404 142 875 474 165 360

 N=41000 Count of occurences of primefactor p in the (partial) Eulerproduct over prime(1)...prime(41000) s\p 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 1 81841 30680 12814 7923 4478 3702 2716 2385 1936 1527 1382 1167 1061 990 919 797 724 692 633 593 561 541 497 466 2 163877 61441 25591 15930 9021 7381 5426 4826 3879 3016 2804 2332 2112 1980 1838 1598 1445 1401 1258 1184 1123 1058 1017 936 3 81841 51149 12814 23840 4478 11087 2716 7196 1936 1527 4205 3503 1061 3025 919 797 724 2095 1897 593 1700 1591 497 466 4 204876 61441 51242 15930 9021 14805 10898 4826 3879 6024 2804 4664 4216 1980 1838 3204 1445 2797 1258 1184 2277 1058 1017 1874 5 81841 30680 23065 7923 22438 3702 2716 2385 1936 1527 7021 1167 5228 990 919 797 724 3468 633 2972 561 541 497 466 6 163877 102440 25591 47837 9021 22149 5426 14425 3879 3016 8478 7014 2112 5992 1838 1598 1445 4175 3786 1184 3421 3171 1017 936 7 81841 30680 12814 14732 4478 3702 2716 2385 1936 10626 1382 1167 1061 6987 919 797 724 692 633 4139 561 541 497 466 8 245875 61441 51242 15930 9021 14805 21766 4826 3879 6024 2804 4664 8412 1980 1838 3204 1445 2797 1258 1184 4560 1058 1017 3738 9 81841 71618 12814 23840 4478 11087 2716 21571 1936 1527 4205 10482 1061 3025 919 797 724 2095 1897 593 5181 1591 497 466 10 163877 61441 46070 15930 45072 7381 5426 4826 3879 3016 14094 2332 10496 1980 1838 1598 1445 6934 1258 5930 1123 1058 1017 936 11 81841 30680 12814 7923 8572 3702 2716 2385 21357 1527 1382 1167 1061 990 919 797 724 692 6881 593 561 541 497 5145 12 204876 102440 51242 47837 9021 44365 10898 14425 3879 6024 8478 14052 4216 5992 1838 3204 1445 8356 3786 1184 6896 3171 1017 1874 relative frequency rounded to integer: count/N*(pk - 1)² 1 2 3 5 7 11 13 17 19 23 29 30 37 41 43 47 53 59 61 67 71 71 80 82 88 2 4 6 10 14 22 26 34 38 46 58 62 74 82 85 95 105 119 123 134 142 142 157 167 177 3 2 5 5 21 11 39 17 57 23 29 92 111 41 130 47 53 59 184 202 71 215 236 82 88 p=1(mod3) 4 5 6 20 14 22 52 68 38 46 115 62 147 165 85 95 211 119 246 134 142 288 157 167 354 5 2 3 9 7 55 13 17 19 23 29 154 37 204 43 47 53 59 305 67 355 71 80 82 88 p=1(mod5) 6 4 10 10 42 22 78 34 114 46 58 186 222 82 258 95 105 119 367 402 142 433 471 167 177 7 2 3 5 13 11 13 17 19 23 203 30 37 41 301 47 53 59 61 67 495 71 80 82 88 p=1(mod7) 8 6 6 20 14 22 52 136 38 46 115 62 147 328 85 95 211 119 246 134 142 577 157 167 706 9 2 7 5 21 11 39 17 170 23 29 92 331 41 130 47 53 59 184 202 71 655 236 82 88 10 4 6 18 14 110 26 34 38 46 58 309 74 410 85 95 105 119 609 134 709 142 157 167 177 11 2 3 5 7 21 13 17 19 252 29 30 37 41 43 47 53 59 61 731 71 71 80 82 972 12 5 10 20 42 22 156 68 114 46 115 186 444 165 258 95 211 119 734 402 142 872 471 167 354