Factorizations of concatenated odd numbers

This is a table of factorizations of “concatenated odd numbers”. These numbers are formed by taking the odd numbers from 1 up to some limit, and concatenating them all together into one number. The first few numbers of this type are: 1, 13, 135, 1357, 13579, 1357911, and 135791113.

After 13579, the numbers in the table are shown with just the first and last odd numbers. The two dots between these numbers represent the odd numbers between 1 and the ending number. For example, 1..11 is short for the number 1357911, and 1..19 is short for 135791113151719.

The period character is used instead of the usual asterisk to mean “multiply”, since it makes the table easier to read.

The last number in a factorization is sometimes written as the letter “p” or “c” followed by a number. This notation means a prime or composite number, respectively, with the given number of digits. So, p108 means a prime number of 108 digits, and c127 means a composite number (whose factors are unknown) of 127 digits. The actual value of this last factor can be computed from the other factors.

Only a few of the following numbers are prime, namely 13, 1..19, 1..31, 1..67, and 1..97. Interestingly, the last odd number that makes up each of these numbers (3, 19, 31, 67, and 97) is itself prime. Surely that cannot always be the case. I have checked up to 1..3001 without finding another prime of this form.

Update: July, 2003: Donovan Johnson reports that 1357...5139 [with 9725 digits] is a probable prime. Assuming that the number is in fact prime, it breaks the above pattern, since 5139=3*3*571 is not prime. He has checked up through 1357...17001 without finding any more primes.

November, 2010: Thanks to Bob Backstrom for completing the factorization of 1357...121.

Please let me know if you find any mistakes in this table, or have found any additional factors, or if you know the next prime number in this series.

First incomplete factorization: 1..135
Last complete factorization: 1..291


13 = PRIME
135 = 3.3.3.5
1357 = 23.59
13579 = 37.367
1..11 = 3.3.3.19.2647
1..13 = 11617.11689
1..15 = 5.7.11.79.446461
1..17 = 3.3.3.3.5179.3236983
1..19 = PRIME
1..21 = 29.43.493333.22073171
1..23 = 3.3.461.327286365754927
1..25 = 5.5.216221921.25120693133
1..27 = 11.1234464665015629202957
1..29 = 3.3.3.3.3.197.28366048996619918599
1..31 = PRIME
1..33 = 17.113.22189.7404677197.43022943781
1..35 = 3.3.3.5.10343.746022014561.1303585172327
1..37 = 7.11.13.19.139.2221001147.23127041886151531
1..39 = 13.293.652331.5465027583669292395042241
1..41 = 3.3.7.7.7.103.505922674244987.8441378043607063
1..43 = 131.2857.880295131.5937472699.69416048357141
1..45 = 5.2797.43746578836459.22195492326077485839803
1..47 = 3.3.3.47119.5461739.14348891.13619533749245871792431
1..49 = 11.463.1319.58561057.345178684922609477476949072671
1..51 = 25471443030907588399109.533111190390538140653339
1..53 = 3.3.3.67.1951.384747239234430989664902129065185474595967
1..55 = 5.349.877.1259573648090166029779.70445419188051915702413
1..57 = 7.233.20177.412629993872800857491740497983214566243420011
1..59 = 3.3.11.23.31.67409.285383420876337494610191221655399983068456973
1..61 = 181.613.2712127111.145036545103.3111320948699685369511466418889
1..63 = 41.280495501404948169538057.1180759888470603874278679399224499
1..65 = 3.3.3.5.317.691.55792246426261.74141386775586839.11101092389700630772223
1..67 = PRIME
1..69 = 19.107.2222452947224330330898392429. p34
1..71 = 3.3.3.3.11.11.11185621649. p53
1..73 = 40351.40720571. p56
1..75 = 5.5.5.139.874031252491240866372373. p42
1..77 = 3.3.103. p70
1..79 = 7.101.4793.6040019.124298588667269. p47
1..81 = 11.61.61.149.4690932325139857791335723910247. p39
1..83 = 3.3.3.3.7.13.17.17.672377417.100371267727703657. p46
1..85 = 5.17.16712741.38481451. p64
1..87 = 57041.168713.1683601.71972038207.126193617039043.2446645577086210153. p23
1..89 = 3.3.3.17.19.29.641.2885611.121146729700181. p56
1..91 = 351269.1724929.1158093253003009. p60
1..93 = 11.29.1319.1481.4421.5136609943. p66
1..95 = 3.3.5.4751.10463.2867276488649.36957209674829. p55
1..97 = PRIME
1..99 = 7.7.7.19.733.46881706797545797693.235772178228405404273367503. p42
1..101 = 3.3.17.127.1672314507390641063351.7425039926893923212334427155181. p41
1..103 = 541.307693.35365991848280939.563130238350583435603778747. p49
1..105 = 5.13.17.12203.352224750976005428272789397. p70
1..107 = 3.3.3.3.1699.29671.50647.579971114031767. p78
1..109 = 7.27779.141471574927.177050917819.152062349013968639602029712750003. p50
1..111 = 11.43.103. p108
1..113 = 3.3. p115
1..115 = 5.11353.4384042561.731558377677487
         .494037607573255102756835714751802129. p54
1..117 = 17.49471679839.157897867009.383780200259727847
         .6837180924125091939361. p59
1..119 = 3.3.17.31607.77323.9823427.3424545511
         .5939721587698386959386127379300893460452951. p54
1..121 = 5933823881.1881525792650786340531660156405228691228676312963. p70 [BB]
1..123 = 67.73.1123.22769.1026773787182367990594312124117
         .1319372421915813012082403279377027536262513. p47
1..125 = 3.3.3.3.5.5.5.5.23.97.521.2671.802007994222433. p105
Thanks to Donovan Johnson for most of the larger factors below.
1..127 = 7.13.275578207.652125467682957797442871
         .45628038116953381054597011031921010514796811219. p56
1..129 = 11.173.54333011.46171495693
         .89887462474561956502870339388563. p86
1..131 = 3.3.41.21836085163289.16089560715966900178517
         .2086483098446372313119419498933. p74
1..133 = 487.3989.54833.51364174392523123
         .176488963313673361080155345168688390157764282899. p71
1..135 = 5.173.9071306691603092023. c127
1..137 = 3.3.7.562956840629.3449997525173183. p123
1..139 = 1757741.16005623. c141
1..141 = 17.461.12211.2751703.520642121.130368561531340841. p117
1..143 = 3.3.3.3.3.3.3.19.52383723898812180753620283591332362501. p118
1..145 = 5.163.71089.3882423140023924384266896189047649610877. c116
1..147 = 36493.346187.15247447.774651571.10897372629130702087
         .4708372549962465653201.20300921417508468234608479
         .2623769579413673242029813095758799551. p38
1..149 = 3.3.479.21997.29346049.6652514843205883. p138
1..151 = 9467.222137.429083.386483301719653
         .7124565726657982320380718878658707. p109
1..153 = 11721781.164664209.801482839.96811309248012157009. c131
1..155 = 3.3.5.7.11.2819.11251.7074233.9682919. p154
1..157 = 13.31.9974659942920748598374970279525554163. p142
1..159 = 1051.2113.2591.8849.374989. c165
1..159 = 1051.2113.2591.8849.374989.3953673080840813611. c147
1..161 = 3.3.3.3.17.72524069.586510118601495360661. p156
1..163 = 97.65955287.94464863.2518787211683410651
         .139627982830029275928679. c131
1..165 = 5.7.251.397.375507056005999. p173
1..167 = 3.3.17.883.416402171.3408197654287.7830413350575367
         .7650295305201657139. p136
1..169 = 61.1162453.10596824309248066777. c173
1..171 = 16787.31961973301062919. c182
1..173 = 3.3.11.72368531. c196
1..175 = 5.5.53.269.121321.5193527.12173075261
         .1284309274900640223346404199. p154
1..177 = 737675727329.4217755511656049327.34284488808198246931271113. c156
1..179 = 3.3.3.3.13.53.659.5905609.7730035771.140678607059689. c176
1..181 = 403914300146928385269996899473. p188
1..183 = 7.58838453.6196541451563614009.72460402304717696453. p173
1..185 = 3.3.5.39877.5470931093.48378578431.177137985655909
         .25968679920105301. c166
1..187 = 2460299. c220
1..189 = 889250669947. c218
1..191 = 3.3.61.101.313.9257.22051.201653.47947426784771
         .952059940013837195284287389. p171
1..193 = 7.127.87383.53145181.456900056729.20057131004974351110231923. c183
1..195 = 5.28019.481813.634178620568459.1303375470151864268061403. p189
1..197 = 3^6.137.2207.45631.55817.1889820912163.410085828209774449
         .4466019311956138621. c175
1..199 = 11.1039.4441939. c234
1..201 = 665868611.10343729587. c229
1..203 = 3.3.77558557256380894971752154965453857327.  c212
1..205 = 5.127.3169.242509733.403894042451.6846129791341. c215
1..207 = 67.229.560246249. c244
1..209 = 3.3.13.159087901786726822963. c237
1..211 = 7.43.197.1636534469. p249
1..213 = 2179.17029.113426407.14597933219. c240
1..215 = 3.3.3.3.5.13523.1418948624793929461. c244
1..217 = 11.11.41.11411. c264
1..219 = 169038271. c266
1..221 = 3.3.7.67.8053.382229.6663551753.139728569070127
         .2230745234560251907478750711587. c210
1..223 = 19.1051.174613.979050837306265664791. c250
1..225 = 5.5.317.108011.214289861.52283527363.692089299744919. p241
1..227 = 3.3.12787195736732057.45148957500678278265676843. c244
1..229 = 34283.3636071.251131943.35961207253. c260
1..231 = 13.60320513377. p281
1..233 = 3.3.3.3.286025556889. c282
1..235 = 5.8483125373.101024077778617859.274882246238481780320696333. c245
1..237 = 83.8263.238509775102877. c281
1..239 = 3.3.7.673.88028657.1426961539823. c280
1..241 = 31.2857.279448847160661.25810138647468175237
         .139707664163107492516204039. c243
1..243 = 11.2137.421093.2317879.21642503. c287
1..245 = 3.3.5.23.1274437.6057979561.1944756125149.45987942586279. p269
1..247 = 173.1215547237.19555591183.106567630994497. p281
1..249 = 7.17.19.29.31. c313
1..251 = 3.3.3.3.3.3.3.281.147212757118873450813.464505919540058324741
         .19777696779050397274825123. c251
1..253 = 16063.1162279.73165667.2627933430437562623.13678266819277063403
         .52723606412577201077407. c247
1..255 = 5.41.102161. c321
1..257 = 3.3. c331
1..259 = 17.17.2273.13580715437701. c316
1..261 = 11.13.43.73.2281.1023401633. c320
1..263 = 3.3.12936733.9418415485887431. p317
 (to be filled in)
1..291 = 271.1931.25321.160911991.217220470140689957
         .2080123654987228021. p329
1..293 = 3.3.953.1321.2663.4993.630325135782799290719503. c348
1..295 = 5.7.7. c386
1..297 = 11909. c388
1..299 = 3.3.19998029. c386
While looking for the next number of this form that is prime, I noticed the following two consecutive numbers with the same smallest factor (967).
1..2899 = 967.509633 . c5236
1..2901 = 967. c5246

E-mail me at jrhowell@ix.netcom.com

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Last Updated: October 14, 2017.