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OEIS sequences needing factors
From OeisWiki
- Please check with corresponding OEIS entry and with factordb.com to make sure number still needed before embarking on a significant effort.
Mersenne Forum
Many of the listed sequences were subject to factoring efforts at Mersenneforum.org, a discussion board for those interested in factorization and primality searches. In particular, there is a thread about sequences that require some factorization in order to be extended, which comes with the comment:
- The following table lists some OEIS entries for which computing further terms is blocked by finding at least one factor of an integer. In some, cases a complete factorization is required, in others only the smallest factor, or any factor.
- The list is unlikely to be exhaustive nor does inclusion or exclusion from the list indicate any kind or importance or mathematical utility. As near as I can tell many of these sequences have no utility beyond their OEIS entry.
- Rows marked with "*" indicate more terms are needed for the initial sequence lines in the corresponding OEIS entry. That is, the OEIS entry has (or should have) the "more" keyword. As above, it is not an indication of the importance of the sequence.
- In some cases it is possible or likely that considerably more ECM effort has been expended than is indicated below.
ECM efforts
Sources
The following list contains some websites that track ECM or other factoring effort, which may not always be reflected on this page, but often are the sources of the ECM effort listed here:
- yoyo@home performs ECM on a wide range of cofactors.
- The most wanted Cunningham numbers can generally be assumed to have t70 ECM performed on them, as well as most base 2 Cunningham cofactors.
- mersennus.net tracks ECM efforts on Fibonacci and Lucas numbers.
- Studio Kamada tracks ECM efforts for near-repdigit-related numbers.
- mersenne.org tracks ECM efforts for Mersenne numbers with prime n
- factordb does not track unsuccessful ECM efforts, but t-level of approximately the size of the largest known prime factor can be assumed at the least, barring their existence being known solely due to other methods (such as algebraic factorization, which may or may not be reflected accurately on factordb's "more information" subtab).
- Some forms with known algebraic factorizations for a subset of their numbers:
- Cunningham and Homogeneous Cunningham numbers (algebraic and Aurifeuillean)
- Fibonacci and Lucas numbers (Aurifeuillean and via other identities)
- Some forms with formulae for their divisors:
- Fermat numbers (Fermat divisors)
- It can be assumed a large amount of ECM has been run on all Fermat cofactors
- Fermat divisors are primarily found through Proth Prime searches
- Fermat numbers (Fermat divisors)
- Some forms with known algebraic factorizations for a subset of their numbers:
The ECM efforts on this wiki page likely include efforts from the above links, and as such should not be backpropagated to those sources as if they had occurred twice. To be on the safe side, the total effort for a given composite number should be estimated as `max(other source, this page)`, unless a high degree of provenance can be established.
T-levels
To simplify communication of ECM progress, many use the "t-level" metric to condense the [<curves>@<B1>, ...] format down to a single number. For instance, the factoring program yafu can be used with the command line arguments -work <t1> and -pretest <t2> to run an optimal number and size of ECM curves on a composite with existing ECM t1 and desired finishing ECM t2.
Here's yafu's explanation of t-levels:
- A note on the “t-level” terminology used in factor(). Something that has received, say, "t30", has had enough ecm curves run on it so that the probability that a factor of size 30 has been missed is exp(-1) (about 37%). Likewise, t35 indicates that factors of size 35 are expected to be missed about 37% of the time (at which point a 30 digit factor would only be expected to be missed ~5% of the time). t-levels are calculated from tabulated data extracted by A. Schindel from GMP-ECM in verbose mode. See also the GMP-ECM README file. I am unaware if t-level is universally accepted terminology or not, but others frequently use it (mersenneforum.org), and it is a handy way to talk about how much a particular number has been tested with ecm.
Conversion of ECM effort from <curves>@<B1> form to t-level form can be performed easily on Mersenneforum.org user WraithX's web page: ecmprobs.html.
Completion Conditions
If any composite on this page is fully factored, it can be assumed that all of the associated sequences need to be examined for the possibility of a new term being added, and the factored composite listed here will need to be updated to the next "blocking" unfactored composite number associated with the sequences. However, certain sequences don't require their associated composite numbers to be fully factored for this occur. These composites have been denoted below with the following tags:
- [semiprimality] - These numbers only need a single non-trivial divisor to be found. If the composite has two known factors, primality tests can then be done on each to determine if the number is semiprime with little computational effort.
- [k-almost primality] - These numbers, like semiprimes, need k-1 prime factors found to be "completed".
- [smallest factor] - These numbers only need their smallest prime factor found, if the known factor is small, it's not difficult to prove it's the smallest factor. However, if the factor is larger, due to the probabilistic nature of ECM, it's difficult to prove the factor is the smallest, even if we can know it is with high probability.
- [specific small factors] - These numbers require some amount of the smallest prime factors of a number be known, but usually provide a mechanism for which some small factors might not be sufficient for "completion" of the factored number, such as the factors being primitive, or factors having a specific modular congruence. This also runs into the same issue of proving a certain factor is the smallest, and sometimes full factorization is less computationally expensive than proving that a factor is the smallest.
Sequences in the OEIS
Cunningham numbers
Cunningham numbers are of the form , which have particular importance in number theory. More generally, b here may be fractional, giving rise to numbers of the form . Further extending this to quadratic irrational b leads to values of Lucas sequences (including Fibonacci, Lucas, and Pell numbers).
Cunningham numbers admit factorization via cyclotomic polynomials , and thus factorization of Cunningham numbers reduces to that of values of the corresponding cyclotomic polynomials.
id size description known ecm effort -------------------------------------------------------------------------------- b = 2 A002587 C330 2^1129+1 or Phi_{2258}(2) t50 [likely more] also needed by A002589, A046798, A046799, A053285, A054992, A057957, A059886, A067718, A069061, A085029, A086257, A274906, A295501, A366602, A366603, A366604 A002185 C248 2^1139+1 or Phi_{2278}(2) t56 [likely more] A283931 C236 * 2^1151+1 or Phi_{2302}(2) t65 [likely more] A226368 C326 * 2^1168+1 or Phi_{2336}(2) t54 [likely more] also needed by A002590, A057940, A274903, A366605, A366606, A366607, A366608 A002184 C337 2^1207-1 or Phi_{1207}(2) t65 [likely more], [specific small factors] also needed by A002588, A005420, A046051, A046800, A046801, A049093, A049094, A053287, A059499, A075708, A085021, A086251, A097406, A108974, A112927, A237043 A112092 C330 2^1208+1 or Phi_{2416}(2) t48 A003260 C297 2^1213-1 or Phi_{1213}(2) t65, also needed by A046932, A055061, A088863, A100730, A181046 A038553 C284 2^1229-1 or Phi_(1229)(2) t65 A016047 C303 2^1237-1 or Phi_(1237)(2) t65, [smallest factor], also needed by A049479, A136030, A186283, A186522, A212953, A215798, A215799, A215806 A250291 C379 2^1259+1 or Phi_{2518}(2) t47 [likely more] A006514 C385 2^1277-1 or Phi_{1277}(2) t71, [semiprimality], also needed by A085724 A215807 C396 2^1327+1 or Phi_{2654}(2) t46 A057953 C212 2^1503-1 or Phi_{1503}(2) t60 [likely more], also needed by A059890, A085033, A274908, A366651, A366652, A366653, A366654 A057936 C248 2^1509+1 or Phi_{3018}(2) t60 [likely more], also needed by A274905, A366655, A366656, A366657, A366658 A345460 C429 * 2^1559+1 or Phi_{3118}(2) t46 [likely more] A379641 C308 2^1647-1 or Phi_{1647}(2) t48 [likely more], also needed by A381493 A359088 C272 * 2^1653-1 or Phi_{1653}(2) t56 [likely more] A229747 C255 2^2266+1 or Phi_{4532}(2) t56 [likely more] A002586 C1156 2^3968+1 or Phi_{7836}(2) t30 [likely more], [smallest factor], also needed by A366609 A073639 C984 * 2^4495-1 or Phi_{4495}(2) t65 [likely more] A347141 C1416 * 2^4703-1 or Phi_{4703}(2) t15 [smallest factor] A096393 C1201 * 2^4844+1 or Phi_{9688}(2) t20 A133485 C1510 2^5099-1 or Phi_{5099}(2) t30 [likely more] A133485 C1527 2^5099+1 or Phi_{10198}(2) t30 [likely more] A219461 C998,C1027,C1084,C1084 * 2^21700-1 or Phi_{21700}(2) t21 A046052 C1133 2^(2^12)+1 or or Phi_{8192}(2) t55, also needed by A050922, A023394, A070592, A321213 A255770 C1221 * 2^(3*2^11)+1 or Phi_{3*2^12}(2) t18, also needed by A255771 A366671 C4880,C9844 2^(3*2^14)+1 or Phi_{3*2^15}(2) [smallest factor] A092558 C35293 2^117239-1 [2^117239+1 is semiprime] [semiprimality] A007117 C315653 * 2^(2^20)+1 or Phi_{2097152}(2) [smallest factor], also needed by A066263, A073936, A092559, A093179, A366648 A199295 C5050446,C10100891 * 8^(8^8)+1 or 2^50331648+1 or Phi_{100663296}(2) [smallest factor] A263686 C694127911065419642 * 2^(2^61-1)-1 or 2^2305843009213693951-1 or Phi_{2305843009213693951}(2) [smallest factor] also needed by A309130 -------------------------------------------------------------------------------- b = 3 A002591 C269 3^703-1 or Phi_{703}(3) t48, also needed by A057952, A057958, A059885, A059891, A074477, A085028, A085034, A129733, A133801, A274909, A295500, A366575, A366576, A366660, A366661, A366662, A366663 A057941 C327 3^709+1 or Phi_{1418}(3) t47, also needed by A074476, A366577, A366578, A366579, A366580 A002592 C330 3^712+1 or Phi_{1424}(3) t47, also needed by A057935, A366664, A366665, A366666, A366667 A143663 C265 3^731-1 or Phi_{731}(3) t60, [specific small factors], also needed by A379642, A381370 A235365 C321 3^769+1 or Phi_{1538}(3) t60, [likely more], [smallest factor], also needed by A272069 A235366 C300 3^797-1 or Phi_{797}(3) t60, [smallest factor], also needed by A218356 A275377 C466 3^(2^10)+1 or Phi_{2048}(3) t45 A113913 C317 * 3^2187+1 or Phi_{4374}(3) t60 A200918 C479894 * (3^1006002-1)/1006003^2 or Phi_{1006002}(3) 2@1000,1@2000,1@5000,1@10000 -------------------------------------------------------------------------------- b = 10 A003021 C300 10^346+1 or Phi_{692}(10) t50, also needed by A057934, A119704, A269503, A344897, A366668, A366669 A001270 C328 10^353-1 or Phi_{353}(10) t60, also needed by A003020, A005422, A046053, A046107, A046412, A046415, A046416, A046417, A046418, A046419, A046420, A046421, A057951, A059892, A061075, A070528, A070529, A081317, A081318, A085035, A095370, A095413, A095414, A095417, A095418, A102146, A102347, A102380, A112505, A147556, A204845, A295503 A176973 C230 10^383-1 or Phi_{383}(10) t60, [specific small factors] A087020 C350 10^428+1 or Phi_{856}(10) t60, also needed by A087021, A087022, A087023, A087024, A087025, A087026. A003060 C336 10^439-1 or Phi_{439}(10) t60, also needed by A007138 A046414 C449 10^467-1 or Phi_{467}(10) t46, [3-almost primality], also needed by A046430, A095415, A268582 A147554 C252 * (10^477-1)/(10^159-1) or Phi_{477}(10) t56, also needed by A110758 A046413 C509 10^509-1 or Phi_{509}(10) t40, [semiprimality], also needed by A196104 A072848 C315 10^528+1 or Phi_{1056}(10) t48 A275381 C473 10^(2^9)+1 or Phi_{1024}(10) t45, A038371 C950 10^(2^10)+1 or Phi_{2048}(10) t35, [smallest factor] A102050 C16385 * 10^(2^14)+1 or Phi_{32768}(10) 200@1e6, [smallest factor] also needed by A185121 A309358 C2097153 10^(2^21)+1 or Phi_{4194304}(10) [semiprimality] A122787 C354295 Phi_{3^12}(10) or Phi_{531441}(10) [specific small factors] A076670 Cbig (10^9)^(10^9)+1 or Phi_{18000000000}(10) -------------------------------------------------------------------------------- other integer b A057939 C301 5^488+1 or Phi_{976}(5) t47, also needed by A074478, A366615, A366616, A366617, A366618 A057956 C260 5^503-1 or Phi_{503}(5) t60, also needed by A059887, A074479, A085030, A295502, A366611, A366612, A366613 A275378 C329 5^512+1 or Phi_{1024}(5) t60 A143665 C364 5^521-1 or Phi_{521}(5) t60, [semiprimality], also needed by A218357 A057938 C258 6^436+1 or Phi_{872}(6) also needed by A274904, A366627, A366628, A366629, A366630 A057955 C273 6^437-1 or Phi_{437}(6) t56, also needed by A059888, A085031, A274907, A366620, A366621, A366622, A366623, A379639 A275379 C777 6^1024+1 or Phi_{2048}(6) t27 A366670 C3126 6^4096+1 or Phi_{8192}(6) [smallest prime] A366582 C208883385 6^268435456+1 or Phi_{536870912}(6) [semiprimality] A057937 C300 7^397+1 or Phi_{794}(7) t47, also needed by A227575, A366636, A366637, A366638, A366639 A057954 C305 7^421-1 or Phi_{421}(7) t47, also needed by A059889, A074249, A085032, A366632, A366633, A366634, A366635 A218358 C315 7^431-1 or Phi_{431}(7) t50, also needed by A379640, A381494 A275380 C861 7^1024+1 or Phi_{2048}(7) t25 A062308 C334 11^326+1 or Phi_{652}(11) t48, also needed by A366686, A366687, A366688, A366689, A366690 A218359 C344 11^331-1 or Phi_{331}(11) t48, also needed by A274910, A366681, A366682, A366683, A366684, A366685, A379644 A275382 C482 11^512+1 or Phi_{1024}(11) t45 A250288 C335 * 12^311-1 or Phi_{311}(12) t60, [semiprimality], also needed by A252170, A366707, A366708, A366709, A366710, A366711, A366718 A366712 C347 12^326+1 or Phi_{652}(12) t47, also needed by A366713, A366714, A366715, A366716, A366720 A275383 C553 12^512+1 or Phi_{1024}(12) t30,4600@11e6,1000@11e7 also needed by A366702, A366719 A302097 C184 13^256+1 or Phi_{512}(13) t57 A218360 C308 13^417-1 or Phi_{417}(13) t48 A302098 C265 14^256+1 or Phi_{512}(14) t56 A324941 C239 17^212+1 or Phi_{424}(17) t56 A218361 C397 17^373-1 or Phi_{373}(17) t46 A128398 Cbig * (18^(17*7563707819165039903)-1)/(18^17-1) or Phi_{7563707819165039903}(18^17) [smallest prime] A218362 C254 19^239-1 or Phi_{239)(19) t56 A218363 C381 23^307-1 or Phi_{307)(23) t46 A218364 C262 29^223-1 or Phi_{223)(29) t56 A128677 C21101 (102^(103^2)+1)/(102^103+1) or Phi_{21218}(102) A006486 C282 139^139-1 or Phi_{139}(139) t56, also needed by A334167, A354226 A007571 C242 149^149+1 or Phi_{298}(149) t56, also needed by A344859, A115973 A177996 C408 192^193+1 or Phi_{386}(192) t46, [specific small factors] A133378 C1379 521^521+1 or Phi_{1042}(521) A298310 C322 * 656811^99+1 or Phi_{198}(656811) t48 A298398 C276 * 2746511^90+1 or Phi_{180}(2746511) t56 -------------------------------------------------------------------------------- fractional b A268511 C220 * 3^445+5^445 t56 A082869 C312 * 3^653-2^653 or Phi_{653}(3, 2) t50, [semiprimality] A122119 C680 * 2^1024+5^1024 or Phi_{2048}(2, 5) 100@10000,440@1e6, [smallest factor] -------------------------------------------------------------------------------- irrational b A246556 C228 Pell(631) t56, [specific small primes] A250292 C271 * Pell(709) t56, [semiprimality] A086598 C262 Lucas(1412) t56, also needed by A086599, A086600 A022307 C276 Fibonacci(1423) 20158@11e7,2000@26e7, also needed by A060385 A001578 C258 Fibonacci(1453) t56, also needed by A060383, A139044 A124132 C263 * Lucas(1501) t56, also needed by A215907, A236264 A072381 C323 Fibonacci(1543) 20158@11e7,2000@26e7, [semiprimality], also needed by A114842, A278637 A099954 C377 * F(1801) [F^R(1801) is semiprime] 20158@11e7,6500@26e7,650@85e7, [semiprimality], also needed by A072381 A330777 C378 * Lucas(1816) 6000@26e7 A085726 C383 Lucas(1831) 20158@11e7,1000@26e7, [semiprimality] A280681 C289 Fibonacci(2253) t56, also needed by A335976 A115101 C387 * Lucas(2602) 7771@43e6 A060320 C271 Fibonacci(2835) t56 A074699 C385 * Lucas(3072) t51 A115051 C457 * Lucas(2342) t46 A115051 C965 * Lucas(4684) 500@25e4,165@1e6
Near powers, factorials, and primorials
id size description known ecm effort -------------------------------------------------------------------------------- near-powers with b = 2 A099441 C523 * 2^1736-1737 t50 [semiprimality] A114970 C339 * 2^1125+1125^2 t54 [semiprimality] A099481 C249 * 2^827-827^2 t56 [semiprimality] A242273 C350 * 2^1152*1152-1 t47 [semiprimality] A242175 C578 * 2^1908*1908+1 t51 [semiprimality], also needed by A242116 A242335 C266 * 4^437*437-1 t56 [semiprimality] A252657 C327 * 4^543-543 t50 [semiprimality] A242204 C333 * 4^547*547+1 t48 [semiprimality] A252789 C461 * 4^765+765 t45 [semiprimality] A252661 C334 * 8^369-369 t48 [semiprimality] A242271 C439 * 8^483*483+1 t46 [semiprimality] A242339 C526 * 8^579*579-1 t40 [semiprimality] A085745 C373 * 2^1239+1239 7771@43e6 [semiprimality] A165767 C449 * 2^1489-1489 t46 [semiprimality], also needed by A165768, A165769 A289117 C249 * 2^817*155+1 t52 [semiprimality] A244609 C367 659*2^1208-1 t47 [smallest factor] A100497 C436 * (2^361+1)^4-2 t46 [semiprimality] A268574 C258 * (2^428+1)^2-2 t56 [semiprimality] A269264 C232 * (2^385-1)^2-2 t56 [semiprimality] A360993 C367 * (2^406-1)^3+2 t47 [semiprimality] A360994 C321 * (2^355+1)^3-2 t50 [semiprimality] A268110 C327 * (2^543-542)*2^543+1 t53 [semiprimality] -------------------------------------------------------------------------------- near-powers with b = 3 A114971 C205 * 3^428+428^3 t56 [semiprimality] A252662 C239 * 9^250-250 t56 [semiprimality] A081715 C246 * 3^514+2 t56 [semiprimality] A252656 C299 * 3^626-626 t56 [semiprimality] A080892 C314 * 3^658-2 t48 [semiprimality] A242274 C417 * 3^866*866-1 t46 [semiprimality] A242203 C430 * 3^894*894+1 t46 [semiprimality] A242340 C492 * 9^512*512-1 t45 [semiprimality] A242272 C555 * 9^578*578+1 t35 [semiprimality] A252788 C669 * 3^1402+1402 904@1e6,450@3e6 [semiprimality] A252794 C827 * 9^866+866 t45 [semiprimality] -------------------------------------------------------------------------------- near-powers with b = 5 A242336 C376 * 5^534*534-1 t47 [semiprimality] A252790 C536 * 5^766+766 t40 [semiprimality] A252658 C568 * 5^812-812 t35 [semiprimality] A114973 C607 * 5^868+868^5 t35 [semiprimality] A242205 C707 * 5^1006*1006+1 t35 [semiprimality] -------------------------------------------------------------------------------- near-powers with b = 6 A252791 C261 * 6^335+335 t46 [semiprimality] A242269 C342 * 6^436*436+1 t48 [semiprimality] A242337 C332 * 6^423*423-1 t48 [semiprimality] A252659 C481 * 6^617-617 t45 [semiprimality] -------------------------------------------------------------------------------- near-powers with b = 7 A252660 C325 * 7^384-384 t47 [semiprimality] A114974 C432 * 7^510+510^7 t46 [semiprimality] A242270 C612 * 7^720*720+1 t35 [semiprimality] A242338 C431 * 7^506*506-1 t46 [semiprimality] -------------------------------------------------------------------------------- near-powers with b = 10 A252795 C218 * 10^217+217 t56 [semiprimality] A252663 C269 * 10^269-269 t45 [semiprimality] A216378 C417 * 10^414*414+1 t46 [semiprimality] A242341 C599 * 10^596*596-1 t35 [semiprimality] A072288 Cbig * 10^(10^100)+2, need factor > 16 A078814 Cbig * 10^(10^100)-7, need factor > 16 -------------------------------------------------------------------------------- near-powers with b > 10 A099497 C414 * 182^183-183^182 t48 [semiprimality] A309747 C561 * 236^236+235^235 t35 [semiprimality] A219978 C237 * 115^115-114^114 t56 [semiprimality] A259026 C398 * 290*23^290-1 t46 [semiprimality] -------------------------------------------------------------------------------- near-factorials A394730 C133 * 8*89!+1 A181186 C187 (2^104-1)*104!+1 t59 A095194 C215 10*127!+1 t56 [semiprimality] A085747 C175 108!+109 t57 [semiprimality] A152089 C155 4*125!+1 t47 [specific small factors], also needed by A180590 A100013 C178 110!+7 t55 A063684 C182 * 118!+2 t55 A002582 C214 136!-1 t56 also needed by A054991, A064145, A093082 A083340 C219 75!^2+1 t56 [semiprimality], also needed by A083341 A286181 C222 * 139!-1 t56 also needed by A286208 A002583 C242 140!+1 t63 also needed by A054990, A064144, A064295, A078778, A181764, A264890 A078781 C272 * 154!-1 t56 [semiprimality], also needed by A080802 A098594 C2356 * 929!+1 4590@11e6 [semiprimality] A096225 C106520655 * 15750503!+1 [smallest factor] -------------------------------------------------------------------------------- near-primorials A369245 C171 * 421#+10 t51 A002585 C196 523#+1 t56 also needed by A054988 A002584 C213 541#-1 t56 also needed by A054989 A065314 C234 576#-577 t56 [smallest factor] A065316 C183 466#-467 t57 A065315 C180 442#+443 t54 [smallest factor], also needed by A065317 A250293 C359 * 859#+1 t47 [semiprimality], also needed by A085725 A250294 C458 * 1091#-1 t46 [semiprimality], also needed by A364840 -------------------------------------------------------------------------------- other near-products of primes A104358 C190 A104357(182) t59 [smallest factor] A104359 C184 A104357(162) t57 also needed by A104360, A104361, A104362, A104363 A104366 C153 A104365(171) t52 [smallest factor] A104367 C174 A104365(159) t54 also needed by A104368, A104369, A104370, A104371 A308078 C217 * binomial(101^2,101)-101^101 t57 A309290 C143 * binomial(97^2,97)-97^2 t52
Recurrence sequences involving factorization
id size description known ecm effort -------------------------------------------------------------------------------- Euclid-Mullin sequences A000945 C335 EuclidMullin52 t58 [smallest factor], also needed by A056756, A051318 A051308 C347 EuclidMullin[5]58 7771@43e6 [smallest factor] A051309 C313 EuclidMullin[11]56 7771@43e6 [smallest factor] A051310 C204 EuclidMullin[13]37 t57 [smallest factor] A051311 C232 EuclidMullin[17]31 t50 [smallest factor] A051312 C284 EuclidMullin[19]51 t56 [smallest factor] A051313 C355 EuclidMullin[23]38 t47 [smallest factor] A051314 C343 EuclidMullin[29]57 t55 [smallest factor] A051315 C240 EuclidMullin[31]38 t56 [smallest factor] A051316 C202 EuclidMullin[37]43 t56 [smallest factor] A051317 C362 EuclidMullin[41]38 t47 [smallest factor] A051319 C194 EuclidMullin[47]36 t59.5 [smallest factor] A051320 C229 EuclidMullin[53]49 t50 [smallest factor] A051321 C258 EuclidMullin[59]49 t56 [smallest factor] A051322 C416 EuclidMullin[61]43 t53 [smallest factor] A051323 C143 EuclidMullin[67]43 t56 [smallest factor] A051324 C186 EuclidMullin[71]45 t56 [smallest factor] A051325 C397 EuclidMullin[73]45 t46 [smallest factor] A051326 C292 EuclidMullin[79]32 t56 [smallest factor] A051327 C296 EuclidMullin[83]65 t56 [smallest factor] A051328 C743 EuclidMullin[89]79 4590@11e6 [smallest factor] A051330 C261 EuclidMullin[97]52 t56 [smallest factor] A051331 C334 EuclidMullin[131071]37 t48 [smallest factor] A051332 C285 EuclidMullin[65537]71 t56 [smallest factor] A051333 C829 EuclidMullin[257]85 t35 [smallest factor] A051334 C328 EuclidMullin[8191]60 4590@11e6 [smallest factor] A051335 C564 EuclidMullin[127]66 4590@11e6 [smallest factor] A093782 C429 * EuclidMullin[8581]31 t46 [smallest factor] A094152 C362 EuclidMullin[32687]51 t50 [smallest factor] A261703 C402 EuclidMullin[139]66 t46 [smallest factor] -------------------------------------------------------------------------------- aliquot sequences A008892 C208 A008892(2157) t56 A014360 C200 A014360(1210) t50 A014361 C194 A014361(3519) t58 A014362 C194 A014362(1101) t53 A014363 C197 A014363(1109) t50 A014364 C181 A014364(2194) t55 A014365 C176 A014365(3839) t55 A152466 C1022 A152466(113)+1 t21 -------------------------------------------------------------------------------- A000946 C332 prod(A000946(k),k=1..14)+1 1000@85e7 A005265 C367 prod(A005265(k),k=1..68)-1 t47 [smallest factor] A005266 C211 prod(A005266(k),k=1..14)-1 t56 A057204 C1314 4*prod(A057204(k),k=1..47)^2+3 1000@1e6 [specific small factors] A057205 C345 4*prod(A057205(k),k=1..24)-1 t48 [specific small factors] A057206 C259 6*prod(A057206(k),k=1..17)-1 t56 [specific small factors] A057207 C572 4*prod(A057207(k),k=1..41)^2+1 7600@43e6 [specific small factors], cf. http://mersenneforum.org/showpost.php?p=334311&postcount=60 A057208 C414 prod(A057208(k),k=1..18)^2+4 t46 [specific small factors] A084599 C211 prod(A084599(n),n=1..14)-1 t56 A102926 C472 prod(A102926(k),k=1..111)-1 2300@11e7 [smallest factor] A102926 C432 prod(A102926(k),k=1..111)+1 2060@11e7 [smallest factor] A124984 C923 prod(A124984(n),n=1..15)^2+2 t25 [specific small factors] A124985 C257 8*prod(A124985(n),n=1..12)^2-1 t56 [specific small factors] A124986 C554 * 4*prod(A124986(n),n=1..14)^2+1 t35 [specific small factors] A124987 C366 * prod(A124987(n),n=1..15)^2+4 t50 [specific small factors] A124988 C1197 * 4*prod(A124988(n),n=1..21)^2+3 t20 [specific small factors] A124989 C853 100*prod(A124989(n),n=1..14)^2-5 t25 [specific small factors] A124990 C174 * Phi_{12}(prod(A124990(n),n=1..8)) t54 [specific small factors] A124991 C928 Phi_{5}(5*prod(A124991(n),n=1..34)) t25 [specific small factors] A124992 C561 Phi_{7}(7*prod(A124992(n),n=1..21)) t35 [specific small factors] A124993 C971 Phi_{11}(11*prod(A124993(n),n=1..14)) t22 [specific small factors] A125037 C2117 Phi_{13}(13*prod(A125037(n),n=1..25)) 1000@1e6 [specific small factors] A125038 C1164 * Phi_{17}(17*prod(A125038(n),n=1..14)) 1000@1e6 [specific small factors] A125039 C745 (2*prod(A125039(n),n=1..29))^4+1 4590@11e6 [specific small factors] A125040 C593 * (2*prod(A125040(n),n=1..10))^8+1 4590@11e6 [specific small factors] A125041 C1056 (2*prod(A125041(n),n=1..20))^4+1 1000@1e6 [specific small factors] A125042 C193 * (2*prod(A125042(n),n=1..4))^8+1 t59.5 [specific small factors] A125043 C1057 * Phi_{9}(3*prod(A125043(n),n=1..21)) 1000@1e6 [specific small factors] A125044 C2958 Phi_{27}(3*prod(A125044(n),n=1..23)) t44.5 [specific small factors] A125045 C347 * prod(A125045(n),n=1..64)+2 t48 [specific small factors] A217759 C534 4*prod(A217759(n),n=1..51)^2-1 890@43e6 [specific small factors] A218467 C276 prod(A218467(n),n=1..19)+1 t56 [specific small factors] -------------------------------------------------------------------------------- A031439 C341 A031439(24)^2+1 17900@11e7 A031440 C199 A031440(23)^2-2 t56 A031442 C187 A151799(A031442(21))*A031442(21)-1 t57 A034970 C522 A034970(34)*A034970(35)-1 t36 A037274 C251 A037276^{118}(49) t61 A048986 C172 A048985^{288}(2295) t60 A062962 C228 * A001697(13) t56 A082021 C195 A151799^{2}(A082021(25))*A082021(25)+2 t60 A082132 C543 A151799(A082132(23))*A082132(23)-2 t35 A096098 C1577 concat(A096098(n),n=1..182) t38 [specific small factors] A120716 C1101 * A037279^{4}(8) 4590@11e6 A130139 C364 * A037279^{6}(45) 17900@11e7 A130140 C36562 * A361320^{7}(15) 100@10000 A130141 C235 * A361580^{4}(35) t56 A130142 C1437 * A361581^{5}(45) t15 A177876 C18701 * A003010(15) A177879 C4686 * A003010(13) [smallest factor] A191648 C3840 * A130846^{4}(7) A195264 C178 * A195265(110) t56 also needed by A195265 A330291 C480 concat(A330291(n),n=1..57)/1352659 t45 [smallest factor]
Other sequences
id size description known ecm effort -------------------------------------------------------------------------------- A046461 C5497 * Sm(1651) 4590@11e6 [semiprimality] A079560 C200 * A005150(19) t56 also needed by A079562 A087552 C684 A065447(37)/1111111111111111111 t47 A091335 C416 * Sylvester(11) 17900@11e7 also needed by A091336 A323605 C785 * Sylvester(12) t40 [smallest factor] A101757 C288 * Tribonacci(1091) t50 [semiprimality] A108728 C216 * A019520(91) t56 also needed by A105388 A109757 C414 * tens_complement_factorial(191)+1 t46 [semiprimality] A109758 C183 * tens_complement_factorial(112)-1 t57 [semiprimality] A113773 C285 * A008352(13) t57 A153357 C207 A001008(476) t53 [semiprimality] A177892 C540 * A003010(10) 17900@11e7 A249909 C310 * Euler(188) t48 [smallest factor] A250295 C263 * A005165(150) t56 [semiprimality] A110760 C205 * A007942(56) t56 also needed by A361624 A110759 C185 * A173426(63) t56 A110757 C183 A000422(110) t57 A113825 C371 * A008351(14) t52 A116087 C180 * A000041(A000045(24)) t57 A078604 C201 A011545(201) t50 also needed by A089282, A089283, A089284, A089285, A089286, A089287, A089288, A089289 A089281 C574 A011545(573) t47 [smallest factor]
