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Denis Roegel. A reconstruction of Kaván’s table of factors (1937). [Research Report] 2011. �hal-
00654432�
Kaván’s table of factors (1937)
Denis Roegel
1 November 2011
Ďala, now Hurbanovo in Slovakia. He retired from the directorship in 1927 and died in 1933.
12 Kaván’s table (1934)
Kaván’s table was first published in Czech in 1934 [9, 1]. According to the preface of the English translation of Kaván’s table published in 1937 [10], the table was started in 1917. During the first year, Kaván worked with Dr. V. Zelinka and Miss B. Zubatá and computed the multiples of all prime numbers from 2 to 2909 and of all their powers.
The checking, printing and proof-reading were done in the years after 1918. The cost of printing the tables was covered by the publication fund of the observatory of Stará Ďala.
Kaván’s table gives the complete decomposition in simple factors of every integer from 0 to 255999. It was apparently the first table giving complete decompositions beyond 100000. Each page covers 500 integers and the whole table spans 512 pages. Each line of a page gives the decomposition of ten integers, in a straightforward way. Numbers which are prime are marked in bold face.
As explained in Arthur Beer’s introduction to the table, the table was checked against Inghirami’s table [7] (probably the 1919 edition [8]) for numbers up to 100000, against Goldberg’s table [6] for numbers up to 251647, and against Chernac’s table [3] for every decomposition.
3 Links with other tables
Peters’ table [13] is very similar to Kaván’s table and was published at almost the same time. But it gives only the complete decompositions up to 100000.
Henry E. Merritt used Kaván’s table for his table of “useful numbers” [12]. Merritt’s table gives the decomposition of numbers whose factors are greater or equal to 7 and lower or equal to 127, from 1 to 200000.
1
A brief biography of Kaván precedes his tables [10].
are marked so. We have added notes about the contents of the articles in certain cases.
[1] Raymond Clare Archibald. Jiří Kaván: Rozklad všetkých čísel celých od 2 do 256000 v prvočinitele (review). Scripta Mathematica, 4:338, 1936.
[2] Maarten Bullynck. Factor tables 1657–1817, with notes on the birth of number theory. Revue d’histoire des mathématiques, 16(2):133–216, 2010.
[3] Ladislaus Chernac. Cribrum arithmeticum sive, tabula continens numeros primos, a compositis segregatos, occurrentes in serie numerorum ab unitate progredientium, usque ad decies centena millia, et ultra haec, ad viginti millia (1020000). Numeris compositis, per 2, 3, 5 non dividuis, adscripti sunt divisores simplices, non minimi tantum, sed omnino omnes. Deventer: J. H. de Lange, 1811. [reconstructed in [14]]
[4] James Whitbread Lee Glaisher. Report of the committee on mathematical tables.
London: Taylor and Francis, 1873. [Also published as part of the “Report of the forty-third meeting of the British Association for the advancement of science,” London: John Murray, 1874.
A review by R. Radau was published in the Bulletin des sciences mathématiques et astronomiques, volume 11, 1876, pp. 7–27]
[5] James Whitbread Lee Glaisher. Table, mathematical. In Hugh Chisholm, editor, The Encyclopædia Britannica, 11th edition, volume 26, pages 325–336. Cambridge, England: at the University Press, 1911.
[6] B. Goldberg. Primzahlen und Faktoren-Tafeln von 1 bis 251647. Leipzig: Nies, 1862. [not seen]
[7] Giovanni Inghirami. Elementi di matematiche, volume 1. Firenze: coi tipi Calasanziani, 1841. [second edition] [The table of factors was reconstructed in [15].]
[8] Giovanni Inghirami. Table des nombres premiers et de la décomposition des nombres de 1 à 100000. Paris: Gauthiers-Villars et Cie, 1919. [The table is supplemented by another table by Ernest Lebon.]
[9] Jiří Kaván. Rozklad všetkých čísel celých od 2 do 256000 v prvočinitele / Tabula omnibus a 2 usque ad 256000 numeris integris omnes divisores primos praebens.
Prague: B. Stýblo, 1934. [not seen, translated in [10]]
2
Note on the titles of the works: Original titles come with many idiosyncrasies and features (line splitting, size, fonts, etc.) which can often not be reproduced in a list of references. It has therefore seemed pointless to capitalize works according to conventions which not only have no relation with the original work, but also do not restore the title entirely. In the following list of references, most title words (except in German) will therefore be left uncapitalized. The names of the authors have also been homogenized and initials expanded, as much as possible.
The reader should keep in mind that this list is not meant as a facsimile of the original works. The
original style information could no doubt have been added as a note, but we have not done it here.
[12] Henry Edward Merritt. Gear trains, including a Brocot table of decimal equivalents and a table of factors of all useful numbers up to 200,000. London: Sir Isaac
Pitman & Sons, Ltd., 1947. [The table of factors is reconstructed in [16].]
[13] Johann Theodor Peters, Alfred Lodge, Elsie Jane Ternouth, and Emma Gifford.
Factor table giving the complete decomposition of all numbers less than 100,000.
London: Office of the British Association, 1935. [introduction by Leslie J. Comrie, and bibliography of tables by James Henderson, reprinted in 1963] [reconstructed in [17]]
[14] Denis Roegel. A reconstruction of Chernac’s Cribrum arithmeticum (1811).
Technical report, LORIA, Nancy, 2011. [This is a reconstruction of [3].]
[15] Denis Roegel. A reconstruction of Inghirami’s table of factors (1841). Technical report, LORIA, Nancy, 2011. [This is a reconstruction of [7].]
[16] Denis Roegel. A reconstruction of Merritt’s table of “useful numbers” (1947).
Technical report, LORIA, 2011. [This is a reconstruction of a table in [12].]
[17] Denis Roegel. A reconstruction of the table of factors of Peters, Lodge, Ternouth, and Gifford (1935). Technical report, LORIA, Nancy, 2011. [This is a recalculation of the tables of [13].]
[18] Paul Peter Heinrich Seelhoff. Geschichte der Factorentafeln. Archiv der
Mathematik und Physik, 70:413–426, 1884.
0 1 2 3 22 5 2.3 7 23 32 0
1 2.5 11 22.3 13 2.7 3.5 24 17 2.32 19 1
2 22.5 3.7 2.11 23 23.3 52 2.13 33 22.7 29 2
3 2.3.5 31 25 3.11 2.17 5.7 22.32 37 2.19 3.13 3
4 23.5 41 2.3.7 43 22.11 32.5 2.23 47 24.3 72 4
5 2.52 3.17 22.13 53 2.33 5.11 23.7 3.19 2.29 59 5
6 22.3.5 61 2.31 32.7 26 5.13 2.3.11 67 22.17 3.23 6
7 2.5.7 71 23.32 73 2.37 3.52 22.19 7.11 2.3.13 79 7
8 24.5 34 2.41 83 22.3.7 5.17 2.43 3.29 23.11 89 8
9 2.32.5 7.13 22.23 3.31 2.47 5.19 25.3 97 2.72 32.11 9
10 22.52 101 2.3.17 103 23.13 3.5.7 2.53 107 22.33 109 10
11 2.5.11 3.37 24.7 113 2.3.19 5.23 22.29 32.13 2.59 7.17 11
12 23.3.5 112 2.61 3.41 22.31 53 2.32.7 127 27 3.43 12
13 2.5.13 131 22.3.11 7.19 2.67 33.5 23.17 137 2.3.23 139 13 14 22.5.7 3.47 2.71 11.13 24.32 5.29 2.73 3.72 22.37 149 14 15 2.3.52 151 23.19 32.17 2.7.11 5.31 22.3.13 157 2.79 3.53 15 16 25.5 7.23 2.34 163 22.41 3.5.11 2.83 167 23.3.7 132 16 17 2.5.17 32.19 22.43 173 2.3.29 52.7 24.11 3.59 2.89 179 17 18 22.32.5 181 2.7.13 3.61 23.23 5.37 2.3.31 11.17 22.47 33.7 18 19 2.5.19 191 26.3 193 2.97 3.5.13 22.72 197 2.32.11 199 19 20 23.52 3.67 2.101 7.29 22.3.17 5.41 2.103 32.23 24.13 11.19 20 21 2.3.5.7 211 22.53 3.71 2.107 5.43 23.33 7.31 2.109 3.73 21 22 22.5.11 13.17 2.3.37 223 25.7 32.52 2.113 227 22.3.19 229 22 23 2.5.23 3.7.11 23.29 233 2.32.13 5.47 22.59 3.79 2.7.17 239 23 24 24.3.5 241 2.112 35 22.61 5.72 2.3.41 13.19 23.31 3.83 24 25 2.53 251 22.32.7 11.23 2.127 3.5.17 28 257 2.3.43 7.37 25 26 22.5.13 32.29 2.131 263 23.3.11 5.53 2.7.19 3.89 22.67 269 26 27 2.33.5 271 24.17 3.7.13 2.137 52.11 22.3.23 277 2.139 32.31 27 28 23.5.7 281 2.3.47 283 22.71 3.5.19 2.11.13 7.41 25.32 172 28 29 2.5.29 3.97 22.73 293 2.3.72 5.59 23.37 33.11 2.149 13.23 29 30 22.3.52 7.43 2.151 3.101 24.19 5.61 2.32.17 307 22.7.11 3.103 30 31 2.5.31 311 23.3.13 313 2.157 32.5.7 22.79 317 2.3.53 11.29 31 32 26.5 3.107 2.7.23 17.19 22.34 52.13 2.163 3.109 23.41 7.47 32 33 2.3.5.11 331 22.83 32.37 2.167 5.67 24.3.7 337 2.132 3.113 33 34 22.5.17 11.31 2.32.19 73 23.43 3.5.23 2.173 347 22.3.29 349 34 35 2.52.7 33.13 25.11 353 2.3.59 5.71 22.89 3.7.17 2.179 359 35 36 23.32.5 192 2.181 3.112 22.7.13 5.73 2.3.61 367 24.23 32.41 36 37 2.5.37 7.53 22.3.31 373 2.11.17 3.53 23.47 13.29 2.33.7 379 37 38 22.5.19 3.127 2.191 383 27.3 5.7.11 2.193 32.43 22.97 389 38 39 2.3.5.13 17.23 23.72 3.131 2.197 5.79 22.32.11 397 2.199 3.7.19 39 40 24.52 401 2.3.67 13.31 22.101 34.5 2.7.29 11.37 23.3.17 409 40 41 2.5.41 3.137 22.103 7.59 2.32.23 5.83 25.13 3.139 2.11.19 419 41 42 22.3.5.7 421 2.211 32.47 23.53 52.17 2.3.71 7.61 22.107 3.11.13 42 43 2.5.43 431 24.33 433 2.7.31 3.5.29 22.109 19.23 2.3.73 439 43 44 23.5.11 32.72 2.13.17 443 22.3.37 5.89 2.223 3.149 26.7 449 44 45 2.32.52 11.41 22.113 3.151 2.227 5.7.13 23.3.19 457 2.229 33.17 45 46 22.5.23 461 2.3.7.11 463 24.29 3.5.31 2.233 467 22.32.13 7.67 46 47 2.5.47 3.157 23.59 11.43 2.3.79 52.19 22.7.17 32.53 2.239 479 47 48 25.3.5 13.37 2.241 3.7.23 22.112 5.97 2.35 487 23.61 3.163 48 49 2.5.72 491 22.3.41 17.29 2.13.19 32.5.11 24.31 7.71 2.3.83 499 49
50 22.53 3.167 2.251 503 23.32.7 5.101 2.11.23 3.132 22.127 509 50 51 2.3.5.17 7.73 29 33.19 2.257 5.103 22.3.43 11.47 2.7.37 3.173 51 52 23.5.13 521 2.32.29 523 22.131 3.52.7 2.263 17.31 24.3.11 232 52 53 2.5.53 32.59 22.7.19 13.41 2.3.89 5.107 23.67 3.179 2.269 72.11 53 54 22.33.5 541 2.271 3.181 25.17 5.109 2.3.7.13 547 22.137 32.61 54 55 2.52.11 19.29 23.3.23 7.79 2.277 3.5.37 22.139 557 2.32.31 13.43 55 56 24.5.7 3.11.17 2.281 563 22.3.47 5.113 2.283 34.7 23.71 569 56 57 2.3.5.19 571 22.11.13 3.191 2.7.41 52.23 26.32 577 2.172 3.193 57 58 22.5.29 7.83 2.3.97 11.53 23.73 32.5.13 2.293 587 22.3.72 19.31 58 59 2.5.59 3.197 24.37 593 2.33.11 5.7.17 22.149 3.199 2.13.23 599 59 60 23.3.52 601 2.7.43 32.67 22.151 5.112 2.3.101 607 25.19 3.7.29 60 61 2.5.61 13.47 22.32.17 613 2.307 3.5.41 23.7.11 617 2.3.103 619 61 62 22.5.31 33.23 2.311 7.89 24.3.13 54 2.313 3.11.19 22.157 17.37 62 63 2.32.5.7 631 23.79 3.211 2.317 5.127 22.3.53 72.13 2.11.29 32.71 63 64 27.5 641 2.3.107 643 22.7.23 3.5.43 2.17.19 647 23.34 11.59 64 65 2.52.13 3.7.31 22.163 653 2.3.109 5.131 24.41 32.73 2.7.47 659 65 66 22.3.5.11 661 2.331 3.13.17 23.83 5.7.19 2.32.37 23.29 22.167 3.223 66 67 2.5.67 11.61 25.3.7 673 2.337 33.52 22.132 677 2.3.113 7.97 67 68 23.5.17 3.227 2.11.31 683 22.32.19 5.137 2.73 3.229 24.43 13.53 68 69 2.3.5.23 691 22.173 32.7.11 2.347 5.139 23.3.29 17.41 2.349 3.233 69 70 22.52.7 701 2.33.13 19.37 26.11 3.5.47 2.353 7.101 22.3.59 709 70 71 2.5.71 32.79 23.89 23.31 2.3.7.17 5.11.13 22.179 3.239 2.359 719 71 72 24.32.5 7.103 2.192 3.241 22.181 52.29 2.3.112 727 23.7.13 36 72 73 2.5.73 17.43 22.3.61 733 2.367 3.5.72 25.23 11.67 2.32.41 739 73 74 22.5.37 3.13.19 2.7.53 743 23.3.31 5.149 2.373 32.83 22.11.17 7.107 74 75 2.3.53 751 24.47 3.251 2.13.29 5.151 22.33.7 757 2.379 3.11.23 75 76 23.5.19 761 2.3.127 7.109 22.191 32.5.17 2.383 13.59 28.3 769 76 77 2.5.7.11 3.257 22.193 773 2.32.43 52.31 23.97 3.7.37 2.389 19.41 77 78 22.3.5.13 11.71 2.17.23 33.29 24.72 5.157 2.3.131 787 22.197 3.263 78 79 2.5.79 7.113 23.32.11 13.61 2.397 3.5.53 22.199 797 2.3.7.19 17.47 79 80 25.52 32.89 2.401 11.73 22.3.67 5.7.23 2.13.31 3.269 23.101 809 80 81 2.34.5 811 22.7.29 3.271 2.11.37 5.163 24.3.17 19.43 2.409 32.7.13 81 82 22.5.41 821 2.3.137 823 23.103 3.52.11 2.7.59 827 22.32.23 829 82 83 2.5.83 3.277 26.13 72.17 2.3.139 5.167 22.11.19 33.31 2.419 839 83 84 23.3.5.7 292 2.421 3.281 22.211 5.132 2.32.47 7.112 24.53 3.283 84 85 2.52.17 23.37 22.3.71 853 2.7.61 32.5.19 23.107 857 2.3.11.13 859 85 86 22.5.43 3.7.41 2.431 863 25.33 5.173 2.433 3.172 22.7.31 11.79 86 87 2.3.5.29 13.67 23.109 32.97 2.19.23 53.7 22.3.73 877 2.439 3.293 87 88 24.5.11 881 2.32.72 883 22.13.17 3.5.59 2.443 887 23.3.37 7.127 88 89 2.5.89 34.11 22.223 19.47 2.3.149 5.179 27.7 3.13.23 2.449 29.31 89 90 22.32.52 17.53 2.11.41 3.7.43 23.113 5.181 2.3.151 907 22.227 32.101 90 91 2.5.7.13 911 24.3.19 11.83 2.457 3.5.61 22.229 7.131 2.33.17 919 91 92 23.5.23 3.307 2.461 13.71 22.3.7.11 52.37 2.463 32.103 25.29 929 92 93 2.3.5.31 72.19 22.233 3.311 2.467 5.11.17 23.32.13 937 2.7.67 3.313 93 94 22.5.47 941 2.3.157 23.41 24.59 33.5.7 2.11.43 947 22.3.79 13.73 94 95 2.52.19 3.317 23.7.17 953 2.32.53 5.191 22.239 3.11.29 2.479 7.137 95 96 26.3.5 312 2.13.37 32.107 22.241 5.193 2.3.7.23 967 23.112 3.17.19 96 97 2.5.97 971 22.35 7.139 2.487 3.52.13 24.61 977 2.3.163 11.89 97 98 22.5.72 32.109 2.491 983 23.3.41 5.197 2.17.29 3.7.47 22.13.19 23.43 98 99 2.32.5.11 991 25.31 3.331 2.7.71 5.199 22.3.83 997 2.499 33.37 99
100 23.53 7.11.13 2.3.167 17.59 22.251 3.5.67 2.503 19.53 24.32.7 1009 100 01 2.5.101 3.337 22.11.23 1013 2.3.132 5.7.29 23.127 32.113 2.509 1019 01 02 22.3.5.17 1021 2.7.73 3.11.31 210 52.41 2.33.19 13.79 22.257 3.73 02 03 2.5.103 1031 23.3.43 1033 2.11.47 32.5.23 22.7.37 17.61 2.3.173 1039 03 04 24.5.13 3.347 2.521 7.149 22.32.29 5.11.19 2.523 3.349 23.131 1049 04 05 2.3.52.7 1051 22.263 34.13 2.17.31 5.211 25.3.11 7.151 2.232 3.353 05 06 22.5.53 1061 2.32.59 1063 23.7.19 3.5.71 2.13.41 11.97 22.3.89 1069 06 07 2.5.107 32.7.17 24.67 29.37 2.3.179 52.43 22.269 3.359 2.72.11 13.83 07 08 23.33.5 23.47 2.541 3.192 22.271 5.7.31 2.3.181 1087 26.17 32.112 08 09 2.5.109 1091 22.3.7.13 1093 2.547 3.5.73 23.137 1097 2.32.61 7.157 09 110 22.52.11 3.367 2.19.29 1103 24.3.23 5.13.17 2.7.79 33.41 22.277 1109 110 11 2.3.5.37 11.101 23.139 3.7.53 2.557 5.223 22.32.31 1117 2.13.43 3.373 11 12 25.5.7 19.59 2.3.11.17 1123 22.281 32.53 2.563 72.23 23.3.47 1129 12 13 2.5.113 3.13.29 22.283 11.103 2.34.7 5.227 24.71 3.379 2.569 17.67 13 14 22.3.5.19 7.163 2.571 32.127 23.11.13 5.229 2.3.191 31.37 22.7.41 3.383 14 15 2.52.23 1151 27.32 1153 2.577 3.5.7.11 22.172 13.89 2.3.193 19.61 15 16 23.5.29 33.43 2.7.83 1163 22.3.97 5.233 2.11.53 3.389 24.73 7.167 16 17 2.32.5.13 1171 22.293 3.17.23 2.587 52.47 23.3.72 11.107 2.19.31 32.131 17 18 22.5.59 1181 2.3.197 7.132 25.37 3.5.79 2.593 1187 22.33.11 29.41 18 19 2.5.7.17 3.397 23.149 1193 2.3.199 5.239 22.13.23 32.7.19 2.599 11.109 19 120 24.3.52 1201 2.601 3.401 22.7.43 5.241 2.32.67 17.71 23.151 3.13.31 120 21 2.5.112 7.173 22.3.101 1213 2.607 35.5 26.19 1217 2.3.7.29 23.53 21 22 22.5.61 3.11.37 2.13.47 1223 23.32.17 52.72 2.613 3.409 22.307 1229 22 23 2.3.5.41 1231 24.7.11 32.137 2.617 5.13.19 22.3.103 1237 2.619 3.7.59 23 24 23.5.31 17.73 2.33.23 11.113 22.311 3.5.83 2.7.89 29.43 25.3.13 1249 24 25 2.54 32.139 22.313 7.179 2.3.11.19 5.251 23.157 3.419 2.17.37 1259 25 26 22.32.5.7 13.97 2.631 3.421 24.79 5.11.23 2.3.211 7.181 22.317 33.47 26 27 2.5.127 31.41 23.3.53 19.67 2.72.13 3.52.17 22.11.29 1277 2.32.71 1279 27 28 28.5 3.7.61 2.641 1283 22.3.107 5.257 2.643 32.11.13 23.7.23 1289 28 29 2.3.5.43 1291 22.17.19 3.431 2.647 5.7.37 24.34 1297 2.11.59 3.433 29 130 22.52.13 1301 2.3.7.31 1303 23.163 32.5.29 2.653 1307 22.3.109 7.11.17 130 31 2.5.131 3.19.23 25.41 13.101 2.32.73 5.263 22.7.47 3.439 2.659 1319 31 32 23.3.5.11 1321 2.661 33.72 22.331 52.53 2.3.13.17 1327 24.83 3.443 32 33 2.5.7.19 113 22.32.37 31.43 2.23.29 3.5.89 23.167 7.191 2.3.223 13.103 33 34 22.5.67 32.149 2.11.61 17.79 26.3.7 5.269 2.673 3.449 22.337 19.71 34 35 2.33.52 7.193 23.132 3.11.41 2.677 5.271 22.3.113 23.59 2.7.97 32.151 35 36 24.5.17 1361 2.3.227 29.47 22.11.31 3.5.7.13 2.683 1367 23.32.19 372 36 37 2.5.137 3.457 22.73 1373 2.3.229 53.11 25.43 34.17 2.13.53 7.197 37 38 22.3.5.23 1381 2.691 3.461 23.173 5.277 2.32.7.11 19.73 22.347 3.463 38 39 2.5.139 13.107 24.3.29 7.199 2.17.41 32.5.31 22.349 11.127 2.3.233 1399 39 140 23.52.7 3.467 2.701 23.61 22.33.13 5.281 2.19.37 3.7.67 27.11 1409 140 41 2.3.5.47 17.83 22.353 32.157 2.7.101 5.283 23.3.59 13.109 2.709 3.11.43 41 42 22.5.71 72.29 2.32.79 1423 24.89 3.52.19 2.23.31 1427 22.3.7.17 1429 42 43 2.5.11.13 33.53 23.179 1433 2.3.239 5.7.41 22.359 3.479 2.719 1439 43 44 25.32.5 11.131 2.7.103 3.13.37 22.192 5.172 2.3.241 1447 23.181 32.7.23 44 45 2.52.29 1451 22.3.112 1453 2.727 3.5.97 24.7.13 31.47 2.36 1459 45 46 22.5.73 3.487 2.17.43 7.11.19 23.3.61 5.293 2.733 32.163 22.367 13.113 46 47 2.3.5.72 1471 26.23 3.491 2.11.67 52.59 22.32.41 7.211 2.739 3.17.29 47 48 23.5.37 1481 2.3.13.19 1483 22.7.53 33.5.11 2.743 1487 24.3.31 1489 48 49 2.5.149 3.7.71 22.373 1493 2.32.83 5.13.23 23.11.17 3.499 2.7.107 1499 49
150 22.3.53 19.79 2.751 32.167 25.47 5.7.43 2.3.251 11.137 22.13.29 3.503 150 51 2.5.151 1511 23.33.7 17.89 2.757 3.5.101 22.379 37.41 2.3.11.23 72.31 51 52 24.5.19 32.132 2.761 1523 22.3.127 52.61 2.7.109 3.509 23.191 11.139 52 53 2.32.5.17 1531 22.383 3.7.73 2.13.59 5.307 29.3 29.53 2.769 34.19 53 54 22.5.7.11 23.67 2.3.257 1543 23.193 3.5.103 2.773 7.13.17 22.32.43 1549 54 55 2.52.31 3.11.47 24.97 1553 2.3.7.37 5.311 22.389 32.173 2.19.41 1559 55 56 23.3.5.13 7.223 2.11.71 3.521 22.17.23 5.313 2.33.29 1567 25.72 3.523 56 57 2.5.157 1571 22.3.131 112.13 2.787 32.52.7 23.197 19.83 2.3.263 1579 57 58 22.5.79 3.17.31 2.7.113 1583 24.32.11 5.317 2.13.61 3.232 22.397 7.227 58 59 2.3.5.53 37.43 23.199 33.59 2.797 5.11.29 22.3.7.19 1597 2.17.47 3.13.41 59 160 26.52 1601 2.32.89 7.229 22.401 3.5.107 2.11.73 1607 23.3.67 1609 160 61 2.5.7.23 32.179 22.13.31 1613 2.3.269 5.17.19 24.101 3.72.11 2.809 1619 61 62 22.34.5 1621 2.811 3.541 23.7.29 53.13 2.3.271 1627 22.11.37 32.181 62 63 2.5.163 7.233 25.3.17 23.71 2.19.43 3.5.109 22.409 1637 2.32.7.13 11.149 63 64 23.5.41 3.547 2.821 31.53 22.3.137 5.7.47 2.823 33.61 24.103 17.97 64 65 2.3.52.11 13.127 22.7.59 3.19.29 2.827 5.331 23.32.23 1657 2.829 3.7.79 65 66 22.5.83 11.151 2.3.277 1663 27.13 32.5.37 2.72.17 1667 22.3.139 1669 66 67 2.5.167 3.557 23.11.19 7.239 2.33.31 52.67 22.419 3.13.43 2.839 23.73 67 68 24.3.5.7 412 2.292 32.11.17 22.421 5.337 2.3.281 7.241 23.211 3.563 68 69 2.5.132 19.89 22.32.47 1693 2.7.112 3.5.113 25.53 1697 2.3.283 1699 69 170 22.52.17 35.7 2.23.37 13.131 23.3.71 5.11.31 2.853 3.569 22.7.61 1709 170 71 2.32.5.19 29.59 24.107 3.571 2.857 5.73 22.3.11.13 17.101 2.859 32.191 71 72 23.5.43 1721 2.3.7.41 1723 22.431 3.52.23 2.863 11.157 26.33 7.13.19 72 73 2.5.173 3.577 22.433 1733 2.3.172 5.347 23.7.31 32.193 2.11.79 37.47 73 74 22.3.5.29 1741 2.13.67 3.7.83 24.109 5.349 2.32.97 1747 22.19.23 3.11.53 74 75 2.53.7 17.103 23.3.73 1753 2.877 33.5.13 22.439 7.251 2.3.293 1759 75 76 25.5.11 3.587 2.881 41.43 22.32.72 5.353 2.883 3.19.31 23.13.17 29.61 76 77 2.3.5.59 7.11.23 22.443 32.197 2.887 52.71 24.3.37 1777 2.7.127 3.593 77 78 22.5.89 13.137 2.34.11 1783 23.223 3.5.7.17 2.19.47 1787 22.3.149 1789 78 79 2.5.179 32.199 28.7 11.163 2.3.13.23 5.359 22.449 3.599 2.29.31 7.257 79 180 23.32.52 1801 2.17.53 3.601 22.11.41 5.192 2.3.7.43 13.139 24.113 33.67 180 81 2.5.181 1811 22.3.151 72.37 2.907 3.5.112 23.227 23.79 2.32.101 17.107 81 82 22.5.7.13 3.607 2.911 1823 25.3.19 52.73 2.11.83 32.7.29 22.457 31.59 82 83 2.3.5.61 1831 23.229 3.13.47 2.7.131 5.367 22.33.17 11.167 2.919 3.613 83 84 24.5.23 7.263 2.3.307 19.97 22.461 32.5.41 2.13.71 1847 23.3.7.11 432 84 85 2.52.37 3.617 22.463 17.109 2.32.103 5.7.53 26.29 3.619 2.929 11.132 85 86 22.3.5.31 1861 2.72.19 34.23 23.233 5.373 2.3.311 1867 22.467 3.7.89 86 87 2.5.11.17 1871 24.32.13 1873 2.937 3.54 22.7.67 1877 2.3.313 1879 87 88 23.5.47 32.11.19 2.941 7.269 22.3.157 5.13.29 2.23.41 3.17.37 25.59 1889 88 89 2.33.5.7 31.61 22.11.43 3.631 2.947 5.379 23.3.79 7.271 2.13.73 32.211 89 190 22.52.19 1901 2.3.317 11.173 24.7.17 3.5.127 2.953 1907 22.32.53 23.83 190 91 2.5.191 3.72.13 23.239 1913 2.3.11.29 5.383 22.479 33.71 2.7.137 19.101 91 92 27.3.5 17.113 2.312 3.641 22.13.37 52.7.11 2.32.107 41.47 23.241 3.643 92 93 2.5.193 1931 22.3.7.23 1933 2.967 32.5.43 24.112 13.149 2.3.17.19 7.277 93 94 22.5.97 3.647 2.971 29.67 23.35 5.389 2.7.139 3.11.59 22.487 1949 94 95 2.3.52.13 1951 25.61 32.7.31 2.977 5.17.23 22.3.163 19.103 2.11.89 3.653 95 96 23.5.72 37.53 2.32.109 13.151 22.491 3.5.131 2.983 7.281 24.3.41 11.179 96 97 2.5.197 33.73 22.17.29 1973 2.3.7.47 52.79 23.13.19 3.659 2.23.43 1979 97 98 22.32.5.11 7.283 2.991 3.661 26.31 5.397 2.3.331 1987 22.7.71 32.13.17 98 99 2.5.199 11.181 23.3.83 1993 2.997 3.5.7.19 22.499 1997 2.33.37 1999 99
200 24.53 3.23.29 2.7.11.13 2003 22.3.167 5.401 2.17.59 32.223 23.251 72.41 200 01 2.3.5.67 2011 22.503 3.11.61 2.19.53 5.13.31 25.32.7 2017 2.1009 3.673 01 02 22.5.101 43.47 2.3.337 7.172 23.11.23 34.52 2.1013 2027 22.3.132 2029 02 03 2.5.7.29 3.677 24.127 19.107 2.32.113 5.11.37 22.509 3.7.97 2.1019 2039 03 04 23.3.5.17 13.157 2.1021 32.227 22.7.73 5.409 2.3.11.31 23.89 211 3.683 04 05 2.52.41 7.293 22.33.19 2053 2.13.79 3.5.137 23.257 112.17 2.3.73 29.71 05 06 22.5.103 32.229 2.1031 2063 24.3.43 5.7.59 2.1033 3.13.53 22.11.47 2069 06 07 2.32.5.23 19.109 23.7.37 3.691 2.17.61 52.83 22.3.173 31.67 2.1039 33.7.11 07 08 25.5.13 2081 2.3.347 2083 22.521 3.5.139 2.7.149 2087 23.32.29 2089 08 09 2.5.11.19 3.17.41 22.523 7.13.23 2.3.349 5.419 24.131 32.233 2.1049 2099 09 210 22.3.52.7 11.191 2.1051 3.701 23.263 5.421 2.34.13 72.43 22.17.31 3.19.37 210 11 2.5.211 2111 26.3.11 2113 2.7.151 32.5.47 22.232 29.73 2.3.353 13.163 11 12 23.5.53 3.7.101 2.1061 11.193 22.32.59 53.17 2.1063 3.709 24.7.19 2129 12 13 2.3.5.71 2131 22.13.41 33.79 2.11.97 5.7.61 23.3.89 2137 2.1069 3.23.31 13 14 22.5.107 2141 2.32.7.17 2143 25.67 3.5.11.13 2.29.37 19.113 22.3.179 7.307 14 15 2.52.43 32.239 23.269 2153 2.3.359 5.431 22.72.11 3.719 2.13.83 17.127 15 16 24.33.5 2161 2.23.47 3.7.103 22.541 5.433 2.3.192 11.197 23.271 32.241 16 17 2.5.7.31 13.167 22.3.181 41.53 2.1087 3.52.29 27.17 7.311 2.32.112 2179 17 18 22.5.109 3.727 2.1091 37.59 23.3.7.13 5.19.23 2.1093 37 22.547 11.199 18 19 2.3.5.73 7.313 24.137 3.17.43 2.1097 5.439 22.32.61 133 2.7.157 3.733 19 220 23.52.11 31.71 2.3.367 2203 22.19.29 32.5.72 2.1103 2207 25.3.23 472 220 21 2.5.13.17 3.11.67 22.7.79 2213 2.33.41 5.443 23.277 3.739 2.1109 7.317 21 22 22.3.5.37 2221 2.11.101 32.13.19 24.139 52.89 2.3.7.53 17.131 22.557 3.743 22 23 2.5.223 23.97 23.32.31 7.11.29 2.1117 3.5.149 22.13.43 2237 2.3.373 2239 23 24 26.5.7 33.83 2.19.59 2243 22.3.11.17 5.449 2.1123 3.7.107 23.281 13.173 24 25 2.32.53 2251 22.563 3.751 2.72.23 5.11.41 24.3.47 37.61 2.1129 32.251 25 26 22.5.113 7.17.19 2.3.13.29 31.73 23.283 3.5.151 2.11.103 2267 22.34.7 2269 26 27 2.5.227 3.757 25.71 2273 2.3.379 52.7.13 22.569 32.11.23 2.17.67 43.53 27 28 23.3.5.19 2281 2.7.163 3.761 22.571 5.457 2.32.127 2287 24.11.13 3.7.109 28 29 2.5.229 29.79 22.3.191 2293 2.31.37 33.5.17 23.7.41 2297 2.3.383 112.19 29 230 22.52.23 3.13.59 2.1151 72.47 28.32 5.461 2.1153 3.769 22.577 2309 230 31 2.3.5.7.11 2311 23.172 32.257 2.13.89 5.463 22.3.193 7.331 2.19.61 3.773 31 32 24.5.29 11.211 2.33.43 23.101 22.7.83 3.52.31 2.1163 13.179 23.3.97 17.137 32 33 2.5.233 32.7.37 22.11.53 2333 2.3.389 5.467 25.73 3.19.41 2.7.167 2339 33 34 22.32.5.13 2341 2.1171 3.11.71 23.293 5.7.67 2.3.17.23 2347 22.587 34.29 34 35 2.52.47 2351 24.3.72 13.181 2.11.107 3.5.157 22.19.31 2357 2.32.131 7.337 35 36 23.5.59 3.787 2.1181 17.139 22.3.197 5.11.43 2.7.132 32.263 26.37 23.103 36 37 2.3.5.79 2371 22.593 3.7.113 2.1187 53.19 23.33.11 2377 2.29.41 3.13.61 37 38 22.5.7.17 2381 2.3.397 2383 24.149 32.5.53 2.1193 7.11.31 22.3.199 2389 38 39 2.5.239 3.797 23.13.23 2393 2.32.7.19 5.479 22.599 3.17.47 2.11.109 2399 39 240 25.3.52 74 2.1201 33.89 22.601 5.13.37 2.3.401 29.83 23.7.43 3.11.73 240 41 2.5.241 2411 22.32.67 19.127 2.17.71 3.5.7.23 24.151 2417 2.3.13.31 41.59 41 42 22.5.112 32.269 2.7.173 2423 23.3.101 52.97 2.1213 3.809 22.607 7.347 42 43 2.35.5 11.13.17 27.19 3.811 2.1217 5.487 22.3.7.29 2437 2.23.53 32.271 43 44 23.5.61 2441 2.3.11.37 7.349 22.13.47 3.5.163 2.1223 2447 24.32.17 31.79 44 45 2.52.72 3.19.43 22.613 11.223 2.3.409 5.491 23.307 33.7.13 2.1229 2459 45 46 22.3.5.41 23.107 2.1231 3.821 25.7.11 5.17.29 2.32.137 2467 22.617 3.823 46 47 2.5.13.19 7.353 23.3.103 2473 2.1237 32.52.11 22.619 2477 2.3.7.59 37.67 47 48 24.5.31 3.827 2.17.73 13.191 22.33.23 5.7.71 2.11.113 3.829 23.311 19.131 48 49 2.3.5.83 47.53 22.7.89 32.277 2.29.43 5.499 26.3.13 11.227 2.1249 3.72.17 49
250 22.54 41.61 2.32.139 2503 23.313 3.5.167 2.7.179 23.109 22.3.11.19 13.193 250 51 2.5.251 34.31 24.157 7.359 2.3.419 5.503 22.17.37 3.839 2.1259 11.229 51 52 23.32.5.7 2521 2.13.97 3.292 22.631 52.101 2.3.421 7.192 25.79 32.281 52 53 2.5.11.23 2531 22.3.211 17.149 2.7.181 3.5.132 23.317 43.59 2.33.47 2539 53 54 22.5.127 3.7.112 2.31.41 2543 24.3.53 5.509 2.19.67 32.283 22.72.13 2549 54 55 2.3.52.17 2551 23.11.29 3.23.37 2.1277 5.7.73 22.32.71 2557 2.1279 3.853 55 56 29.5 13.197 2.3.7.61 11.233 22.641 33.5.19 2.1283 17.151 23.3.107 7.367 56 57 2.5.257 3.857 22.643 31.83 2.32.11.13 52.103 24.7.23 3.859 2.1289 2579 57 58 22.3.5.43 29.89 2.1291 32.7.41 23.17.19 5.11.47 2.3.431 13.199 22.647 3.863 58 59 2.5.7.37 2591 25.34 2593 2.1297 3.5.173 22.11.59 72.53 2.3.433 23.113 59 260 23.52.13 32.172 2.1301 19.137 22.3.7.31 5.521 2.1303 3.11.79 24.163 2609 260 61 2.32.5.29 7.373 22.653 3.13.67 2.1307 5.523 23.3.109 2617 2.7.11.17 33.97 61 62 22.5.131 2621 2.3.19.23 43.61 26.41 3.53.7 2.13.101 37.71 22.32.73 11.239 62 63 2.5.263 3.877 23.7.47 2633 2.3.439 5.17.31 22.659 32.293 2.1319 7.13.29 63 64 24.3.5.11 19.139 2.1321 3.881 22.661 5.232 2.33.72 2647 23.331 3.883 64 65 2.52.53 11.241 22.3.13.17 7.379 2.1327 32.5.59 25.83 2657 2.3.443 2659 65 66 22.5.7.19 3.887 2.113 2663 23.32.37 5.13.41 2.31.43 3.7.127 22.23.29 17.157 66 67 2.3.5.89 2671 24.167 35.11 2.7.191 52.107 22.3.223 2677 2.13.103 3.19.47 67 68 23.5.67 7.383 2.32.149 2683 22.11.61 3.5.179 2.17.79 2687 27.3.7 2689 68 69 2.5.269 32.13.23 22.673 2693 2.3.449 5.72.11 23.337 3.29.31 2.19.71 2699 69 270 22.33.52 37.73 2.7.193 3.17.53 24.132 5.541 2.3.11.41 2707 22.677 32.7.43 270 71 2.5.271 2711 23.3.113 2713 2.23.59 3.5.181 22.7.97 11.13.19 2.32.151 2719 71 72 25.5.17 3.907 2.1361 7.389 22.3.227 52.109 2.29.47 33.101 23.11.31 2729 72 73 2.3.5.7.13 2731 22.683 3.911 2.1367 5.547 24.32.19 7.17.23 2.372 3.11.83 73 74 22.5.137 2741 2.3.457 13.211 23.73 32.5.61 2.1373 41.67 22.3.229 2749 74 75 2.53.11 3.7.131 26.43 2753 2.34.17 5.19.29 22.13.53 3.919 2.7.197 31.89 75 76 23.3.5.23 11.251 2.1381 32.307 22.691 5.7.79 2.3.461 2767 24.173 3.13.71 76 77 2.5.277 17.163 22.32.7.11 47.59 2.19.73 3.52.37 23.347 2777 2.3.463 7.397 77 78 22.5.139 33.103 2.13.107 112.23 25.3.29 5.557 2.7.199 3.929 22.17.41 2789 78 79 2.32.5.31 2791 23.349 3.72.19 2.11.127 5.13.43 22.3.233 2797 2.1399 32.311 79 280 24.52.7 2801 2.3.467 2803 22.701 3.5.11.17 2.23.61 7.401 23.33.13 532 280 81 2.5.281 3.937 22.19.37 29.97 2.3.7.67 5.563 28.11 32.313 2.1409 2819 81 82 22.3.5.47 7.13.31 2.17.83 3.941 23.353 52.113 2.32.157 11.257 22.7.101 3.23.41 82 83 2.5.283 19.149 24.3.59 2833 2.13.109 34.5.7 22.709 2837 2.3.11.43 17.167 83 84 23.5.71 3.947 2.72.29 2843 22.32.79 5.569 2.1423 3.13.73 25.89 7.11.37 84 85 2.3.52.19 2851 22.23.31 32.317 2.1427 5.571 23.3.7.17 2857 2.1429 3.953 85 86 22.5.11.13 2861 2.33.53 7.409 24.179 3.5.191 2.1433 47.61 22.3.239 19.151 86 87 2.5.7.41 32.11.29 23.359 132.17 2.3.479 53.23 22.719 3.7.137 2.1439 2879 87 88 26.32.5 43.67 2.11.131 3.312 22.7.103 5.577 2.3.13.37 2887 23.192 33.107 88 89 2.5.172 72.59 22.3.241 11.263 2.1447 3.5.193 24.181 2897 2.32.7.23 13.223 89 290 22.52.29 3.967 2.1451 2903 23.3.112 5.7.83 2.1453 32.17.19 22.727 2909 290 91 2.3.5.97 41.71 25.7.13 3.971 2.31.47 5.11.53 22.36 2917 2.1459 3.7.139 91 92 23.5.73 23.127 2.3.487 37.79 22.17.43 32.52.13 2.7.11.19 2927 24.3.61 29.101 92 93 2.5.293 3.977 22.733 7.419 2.32.163 5.587 23.367 3.11.89 2.13.113 2939 93 94 22.3.5.72 17.173 2.1471 33.109 27.23 5.19.31 2.3.491 7.421 22.11.67 3.983 94 95 2.52.59 13.227 23.32.41 2953 2.7.211 3.5.197 22.739 2957 2.3.17.29 11.269 95 96 24.5.37 32.7.47 2.1481 2963 22.3.13.19 5.593 2.1483 3.23.43 23.7.53 2969 96 97 2.33.5.11 2971 22.743 3.991 2.1487 52.7.17 25.3.31 13.229 2.1489 32.331 97 98 22.5.149 11.271 2.3.7.71 19.157 23.373 3.5.199 2.1493 29.103 22.32.83 72.61 98 99 2.5.13.23 3.997 24.11.17 41.73 2.3.499 5.599 22.7.107 34.37 2.1499 2999 99
300 23.3.53 3001 2.19.79 3.7.11.13 22.751 5.601 2.32.167 31.97 26.47 3.17.59 300 01 2.5.7.43 3011 22.3.251 23.131 2.11.137 32.5.67 23.13.29 7.431 2.3.503 3019 01 02 22.5.151 3.19.53 2.1511 3023 24.33.7 52.112 2.17.89 3.1009 22.757 13.233 02 03 2.3.5.101 7.433 23.379 32.337 2.37.41 5.607 22.3.11.23 3037 2.72.31 3.1013 03 04 25.5.19 3041 2.32.132 17.179 22.761 3.5.7.29 2.1523 11.277 23.3.127 3049 04 05 2.52.61 33.113 22.7.109 43.71 2.3.509 5.13.47 24.191 3.1019 2.11.139 7.19.23 05 06 22.32.5.17 3061 2.1531 3.1021 23.383 5.613 2.3.7.73 3067 22.13.59 32.11.31 06 07 2.5.307 37.83 210.3 7.439 2.29.53 3.52.41 22.769 17.181 2.34.19 3079 07 08 23.5.7.11 3.13.79 2.23.67 3083 22.3.257 5.617 2.1543 32.73 24.193 3089 08 09 2.3.5.103 11.281 22.773 3.1031 2.7.13.17 5.619 23.32.43 19.163 2.1549 3.1033 09 310 22.52.31 7.443 2.3.11.47 29.107 25.97 33.5.23 2.1553 13.239 22.3.7.37 3109 310 11 2.5.311 3.17.61 23.389 11.283 2.32.173 5.7.89 22.19.41 3.1039 2.1559 3119 11 12 24.3.5.13 3121 2.7.223 32.347 22.11.71 55 2.3.521 53.59 23.17.23 3.7.149 12 13 2.5.313 31.101 22.33.29 13.241 2.1567 3.5.11.19 26.72 3137 2.3.523 43.73 13 14 22.5.157 32.349 2.1571 7.449 23.3.131 5.17.37 2.112.13 3.1049 22.787 47.67 14 15 2.32.52.7 23.137 24.197 3.1051 2.19.83 5.631 22.3.263 7.11.41 2.1579 35.13 15 16 23.5.79 29.109 2.3.17.31 3163 22.7.113 3.5.211 2.1583 3167 25.32.11 3169 16 17 2.5.317 3.7.151 22.13.61 19.167 2.3.232 52.127 23.397 32.353 2.7.227 11.172 17 18 22.3.5.53 3181 2.37.43 3.1061 24.199 5.72.13 2.33.59 3187 22.797 3.1063 18 19 2.5.11.29 3191 23.3.7.19 31.103 2.1597 32.5.71 22.17.47 23.139 2.3.13.41 7.457 19 320 27.52 3.11.97 2.1601 3203 22.32.89 5.641 2.7.229 3.1069 23.401 3209 320 21 2.3.5.107 132.19 22.11.73 33.7.17 2.1607 5.643 24.3.67 3217 2.1609 3.29.37 21 22 22.5.7.23 3221 2.32.179 11.293 23.13.31 3.52.43 2.1613 7.461 22.3.269 3229 22 23 2.5.17.19 32.359 25.101 53.61 2.3.72.11 5.647 22.809 3.13.83 2.1619 41.79 23 24 23.34.5 7.463 2.1621 3.23.47 22.811 5.11.59 2.3.541 17.191 24.7.29 32.192 24 25 2.53.13 3251 22.3.271 3253 2.1627 3.5.7.31 23.11.37 3257 2.32.181 3259 25 26 22.5.163 3.1087 2.7.233 13.251 26.3.17 5.653 2.23.71 33.112 22.19.43 7.467 26 27 2.3.5.109 3271 23.409 3.1091 2.1637 52.131 22.32.7.13 29.113 2.11.149 3.1093 27 28 24.5.41 17.193 2.3.547 72.67 22.821 32.5.73 2.31.53 19.173 23.3.137 11.13.23 28 29 2.5.7.47 3.1097 22.823 37.89 2.33.61 5.659 25.103 3.7.157 2.17.97 3299 29 330 22.3.52.11 3301 2.13.127 32.367 23.7.59 5.661 2.3.19.29 3307 22.827 3.1103 330 31 2.5.331 7.11.43 24.32.23 3313 2.1657 3.5.13.17 22.829 31.107 2.3.7.79 3319 31 32 23.5.83 34.41 2.11.151 3323 22.3.277 52.7.19 2.1663 3.1109 28.13 3329 32 33 2.32.5.37 3331 22.72.17 3.11.101 2.1667 5.23.29 23.3.139 47.71 2.1669 32.7.53 33 34 22.5.167 13.257 2.3.557 3343 24.11.19 3.5.223 2.7.239 3347 22.33.31 17.197 34 35 2.52.67 3.1117 23.419 7.479 2.3.13.43 5.11.61 22.839 32.373 2.23.73 3359 35 36 25.3.5.7 3361 2.412 3.19.59 22.292 5.673 2.32.11.17 7.13.37 23.421 3.1123 36 37 2.5.337 3371 22.3.281 3373 2.7.241 33.53 24.211 11.307 2.3.563 31.109 37 38 22.5.132 3.72.23 2.19.89 17.199 23.32.47 5.677 2.1693 3.1129 22.7.112 3389 38 39 2.3.5.113 3391 26.53 32.13.29 2.1697 5.7.97 22.3.283 43.79 2.1699 3.11.103 39 340 23.52.17 19.179 2.35.7 41.83 22.23.37 3.5.227 2.13.131 3407 24.3.71 7.487 340 41 2.5.11.31 32.379 22.853 3413 2.3.569 5.683 23.7.61 3.17.67 2.1709 13.263 41 42 22.32.5.19 11.311 2.29.59 3.7.163 25.107 52.137 2.3.571 23.149 22.857 33.127 42 43 2.5.73 47.73 23.3.11.13 3433 2.17.101 3.5.229 22.859 7.491 2.32.191 19.181 43 44 24.5.43 3.31.37 2.1721 11.313 22.3.7.41 5.13.53 2.1723 32.383 23.431 3449 44 45 2.3.52.23 7.17.29 22.863 3.1151 2.11.157 5.691 27.33 3457 2.7.13.19 3.1153 45 46 22.5.173 3461 2.3.577 3463 23.433 32.5.7.11 2.1733 3467 22.3.172 3469 46 47 2.5.347 3.13.89 24.7.31 23.151 2.32.193 52.139 22.11.79 3.19.61 2.37.47 72.71 47 48 23.3.5.29 592 2.1741 34.43 22.13.67 5.17.41 2.3.7.83 11.317 25.109 3.1163 48 49 2.5.349 3491 22.32.97 7.499 2.1747 3.5.233 23.19.23 13.269 2.3.11.53 3499 49
