A294492 Numbers m that set records for the ratio A045763(n)/n. Michael Thomas De Vlieger, St. Louis, Missouri 201710312230, revised 201711011530. These numbers have an increasing proportion of nondivisors in the cototient A045763(n) with respect to n. Example: 6 has the nondivisor 4 in the cototient, thus 1/6, while 7 has none, 8 has one (6), 9 has one (6), but 10 has the nondivisors (4,6,8) in the cototient, thus 3/10. Since 3/10 > 1/6, 10 is the next number in the sequence. n = index. m = number that sets a record for the ratio A045763(n)/n. A045763(m) = number of nondivisors in the cototient of m. MN(m) = multiplicity notation of m: these are the exponents of the smallest primes that when multipled produce m. Example: "1 2 0 1" = 2^1 * 3^2 * 5^0 * 7^1 = 126. See below for observations and conjectures. n m A045763(m) MN(m) ----------------------------------------------------------------------------------------------------- 1 1 0 0 2 6 1 1 1 3 10 3 1 0 1 4 14 5 1 0 0 1 5 18 7 1 2 6 22 9 1 0 0 0 1 7 26 11 1 0 0 0 0 1 8 30 15 1 1 1 9 42 23 1 1 0 1 10 60 33 2 1 1 11 66 39 1 1 0 0 1 12 78 47 1 1 0 0 0 1 13 90 55 1 2 1 14 102 63 1 1 0 0 0 0 1 15 114 71 1 1 0 0 0 0 0 1 16 126 79 1 2 0 1 17 138 87 1 1 0 0 0 0 0 0 1 18 150 99 1 1 2 19 210 147 1 1 1 1 20 330 235 1 1 1 0 1 21 390 279 1 1 1 0 0 1 22 420 301 2 1 1 1 23 510 367 1 1 1 0 0 0 1 24 570 411 1 1 1 0 0 0 0 1 25 630 463 1 2 1 1 26 1050 787 1 1 2 1 27 1470 1111 1 1 1 2 28 2310 1799 1 1 1 1 1 29 4620 3613 2 1 1 1 1 30 6930 5443 1 2 1 1 1 31 11550 9103 1 1 2 1 1 32 16170 12763 1 1 1 2 1 33 25410 20083 1 1 1 1 2 34 30030 24207 1 1 1 1 1 1 35 60060 48445 2 1 1 1 1 1 36 90090 72715 1 2 1 1 1 1 37 150150 121255 1 1 2 1 1 1 38 210210 169795 1 1 1 2 1 1 39 330330 266875 1 1 1 1 2 1 40 390390 315415 1 1 1 1 1 2 41 510510 418223 1 1 1 1 1 1 1 42 1021020 836509 2 1 1 1 1 1 1 43 1531530 1254859 1 2 1 1 1 1 1 44 2552550 2091559 1 1 2 1 1 1 1 45 3573570 2928259 1 1 1 2 1 1 1 46 5615610 4601659 1 1 1 1 2 1 1 47 6636630 5438359 1 1 1 1 1 2 1 48 8678670 7111759 1 1 1 1 1 1 2 49 9699690 8040555 1 1 1 1 1 1 1 1 50 19399380 16081237 2 1 1 1 1 1 1 1 51 29099070 24122047 1 2 1 1 1 1 1 1 52 48498450 40203667 1 1 2 1 1 1 1 1 53* 67897830 56285287 1 1 1 2 1 1 1 1 54 106696590 88448527 1 1 1 1 2 1 1 1 55 126095970 104530147 1 1 1 1 1 2 1 1 56 164894730 136693387 1 1 1 1 1 1 2 1 57 184294110 152775007 1 1 1 1 1 1 1 2 58 223092870 186596999 1 1 1 1 1 1 1 1 1 59 446185740 373194253 2 1 1 1 1 1 1 1 1 60 669278610 559791763 1 2 1 1 1 1 1 1 1 61 1115464350 932986783 1 1 2 1 1 1 1 1 1 62 1561650090 1306181803 1 1 1 2 1 1 1 1 1 63 2454021570 2052571843 1 1 1 1 2 1 1 1 1 64 2900207310 2425766863 1 1 1 1 1 2 1 1 1 65 3792578790 3172156903 1 1 1 1 1 1 2 1 1 66 4238764530 3545351923 1 1 1 1 1 1 1 2 1 67 5131136010 4291741963 1 1 1 1 1 1 1 1 2 68 6469693230 5447822127 1 1 1 1 1 1 1 1 1 1 69 12939386460 10895644765 2 1 1 1 1 1 1 1 1 1 70 19409079690 16343467915 1 2 1 1 1 1 1 1 1 1 71 32348466150 27239114215 1 1 2 1 1 1 1 1 1 1 72 45287852610 38134760515 1 1 1 2 1 1 1 1 1 1 73 71166625530 59926053115 1 1 1 1 2 1 1 1 1 1 74 84106011990 70821699415 1 1 1 1 1 2 1 1 1 1 75 109984784910 92612992015 1 1 1 1 1 1 2 1 1 1 76 122924171370 103508638315 1 1 1 1 1 1 1 2 1 1 77 148802944290 125299930915 1 1 1 1 1 1 1 1 2 1 78 187621103670 157986869815 1 1 1 1 1 1 1 1 1 2 79 200560490130 169904385683 1 1 1 1 1 1 1 1 1 1 1 80 401120980260 339808772389 2 1 1 1 1 1 1 1 1 1 1 81 601681470390 509713160119 1 2 1 1 1 1 1 1 1 1 1 82 1002802450650 849521935579 1 1 2 1 1 1 1 1 1 1 1 83 1403923430910 1189330711039 1 1 1 2 1 1 1 1 1 1 1 84 2206165391430 1868948261959 1 1 1 1 2 1 1 1 1 1 1 85 2607286371690 2208757037419 1 1 1 1 1 2 1 1 1 1 1 86 3409528332210 2888374588339 1 1 1 1 1 1 2 1 1 1 1 87 3810649312470 3228183363799 1 1 1 1 1 1 1 2 1 1 1 88 4612891272990 3907800914719 1 1 1 1 1 1 1 1 2 1 1 89 5816254213770 4927227241099 1 1 1 1 1 1 1 1 1 2 1 90 6217375194030 5267036016559 1 1 1 1 1 1 1 1 1 1 2 91 7420738134810 6317118444315 1 1 1 1 1 1 1 1 1 1 1 1 92 14841476269620 12634236890677 2 1 1 1 1 1 1 1 1 1 1 1 93 22262214404430 18951355339087 1 2 1 1 1 1 1 1 1 1 1 1 94 37103690674050 31585592235907 1 1 2 1 1 1 1 1 1 1 1 1 95 51945166943670 44219829132727 1 1 1 2 1 1 1 1 1 1 1 1 96 81628119482910 69488302926367 1 1 1 1 2 1 1 1 1 1 1 1 97 96469595752530 82122539823187 1 1 1 1 1 2 1 1 1 1 1 1 98 126152548291770 107391013616827 1 1 1 1 1 1 2 1 1 1 1 1 99 140994024561390 120025250513647 1 1 1 1 1 1 1 2 1 1 1 1 100 170676977100630 145293724307287 1 1 1 1 1 1 1 1 2 1 1 1 101 215201405909490 183196434997747 1 1 1 1 1 1 1 1 1 2 1 1 102 230042882179110 195830671894567 1 1 1 1 1 1 1 1 1 1 2 1 103 274567310987970 233733382585027 1 1 1 1 1 1 1 1 1 1 1 2 104 304250263527210 260105476063019 1 1 1 1 1 1 1 1 1 1 1 1 1 105 608500527054420 520210952130133 2 1 1 1 1 1 1 1 1 1 1 1 1 106 912750790581630 780316428201343 1 2 1 1 1 1 1 1 1 1 1 1 1 107 1521251317636050 1300527380343763 1 1 2 1 1 1 1 1 1 1 1 1 1 108 2129751844690470 1820738332486183 1 1 1 2 1 1 1 1 1 1 1 1 1 109 3346752898799310 2861160236771023 1 1 1 1 2 1 1 1 1 1 1 1 1 110 3955253425853730 3381371188913443 1 1 1 1 1 2 1 1 1 1 1 1 1 111 5172254479962570 4421793093198283 1 1 1 1 1 1 2 1 1 1 1 1 1 112 5780755007016990 4942004045340703 1 1 1 1 1 1 1 2 1 1 1 1 1 113 6997756061125830 5982425949625543 1 1 1 1 1 1 1 1 2 1 1 1 1 114 8823257642289090 7543058806052803 1 1 1 1 1 1 1 1 1 2 1 1 1 115 9431758169343510 8063269758195223 1 1 1 1 1 1 1 1 1 1 2 1 1 116 11257259750506770 9623902614622483 1 1 1 1 1 1 1 1 1 1 1 2 1 117 12474260804615610 10664324518907323 1 1 1 1 1 1 1 1 1 1 1 1 2 118 13082761331670030 11228680258501647 1 1 1 1 1 1 1 1 1 1 1 1 1 1 119 26165522663340060 22457360517011485 2 1 1 1 1 1 1 1 1 1 1 1 1 1 120 39248283995010090 33686040775529515 1 2 1 1 1 1 1 1 1 1 1 1 1 1 121 65413806658350150 56143401292565575 1 1 2 1 1 1 1 1 1 1 1 1 1 1 122 91579329321690210 78600761809601635 1 1 1 2 1 1 1 1 1 1 1 1 1 1 123 143910374648370330 123515482843673755 1 1 1 1 2 1 1 1 1 1 1 1 1 1 124 170075897311710390 145972843360709815 1 1 1 1 1 2 1 1 1 1 1 1 1 1 125 222406942638390510 190887564394781935 1 1 1 1 1 1 2 1 1 1 1 1 1 1 126 248572465301730570 213344924911817995 1 1 1 1 1 1 1 2 1 1 1 1 1 1 127 300903510628410690 258259645945890115 1 1 1 1 1 1 1 1 2 1 1 1 1 1 128 379400078618430870 325631727496998295 1 1 1 1 1 1 1 1 1 2 1 1 1 1 129 405565601281770930 348089088014034355 1 1 1 1 1 1 1 1 1 1 2 1 1 1 130 484062169271791110 415461169565142535 1 1 1 1 1 1 1 1 1 1 1 2 1 1 131 536393214598471230 460375890599214655 1 1 1 1 1 1 1 1 1 1 1 1 2 1 132 562558737261811290 482833251116250715 1 1 1 1 1 1 1 1 1 1 1 1 1 2 133 614889782588491410 529602053223466643 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 134 1229779565176982820 1059204106446949669 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 135 1844669347765474230 1588806159670449079 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 136 3074448912942457050 2648010266117447899 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 137 4304228478119439870 3707214372564446719 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 138 6763787608473405510 5825622585458444359 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 139 7993567173650388330 6884826691905443179 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 140 10453126304004353970 9003234904799440819 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 141 11682905869181336790 10062439011246439639 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 142 14142464999535302430 12180847224140437279 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 143 17831803695066250890 15358459543481433739 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 144 19061583260243233710 16417663649928432559 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 145 22750921955774182170 19595275969269429019 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 146 25210481086128147810 21713684182163426659 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 147 26440260651305130630 22772888288610425479 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 148 28899819781659096270 24891296501504423119 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 149 32589158477190044730 28154196550210395195 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 150 65178316954380089460 56308393100420823157 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 151 97767475431570134190 84462589650631283887 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 152 162945792385950223650 140770982751052205347 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 153 228124109340330313110 197079375851473126807 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 154 358480743249090492030 309696162052314969727 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 155 423659060203470581490 366004555152735891187 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 156 554015694112230760410 478621341353577734107 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 157 619194011066610849870 534929734453998655567 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 158 749550644975371028790 647546520654840498487 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 159 945085595838511297170 816471699956103262867 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 160 1010263912792891386630 872780093056524184327 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 161 1205798863656031655010 1041705272357786948707 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 162 1336155497564791833930 1154322058558628791627 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 163 1401333814519171923390 1210630451659049713087 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 164 1531690448427932102310 1323247237859891556007 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 165 1727225399291072370690 1492172417161154320387 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 166 1922760350154212639070 1665532558389396635999 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 167 3845520700308425278140 3331065116778793337533 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 168 5768281050462637917210 4996597675168190104603 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 169 9613801750771063195350 8327662791946983638743 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 170 13459322451079488473490 11658727908725777172883 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 171 21150363851696339029770 18320858142283364241163 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 172 24995884552004764307910 21651923259062157775303 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 173 32686925952621614864190 28314053492619744843583 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 174 36532446652930040142330 31645118609398538377723 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 175 44223488053546890698610 38307248842956125446003 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 176 55760050154472166533030 48300444193292506048423 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 177 59605570854780591811170 51631509310071299582563 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 178 71142132955705867645590 61624704660407680184983 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 179 78833174356322718201870 68286834893965267253263 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 180 82678695056631143480010 71617900010744060787403 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 181 90369736457247994036290 78280030244301647855683 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 182 101906298558173269870710 88273225594638028458103 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 183 113442860659098545705130 98266420944974409060523 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 184 117288381359406970983270 101854713853518018401127 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 185 234576762718813941966540 203709427707036036933325 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 186 351865144078220912949810 305564141560554055596595 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 187 586441906797034854916350 509273569267590092923135 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 188 821018669515848796882890 712982996974626130249675 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 189 1290172194953476680815970 1120401852388698204902755 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 190 1524748957672290622782510 1324111280095734242229295 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 191 1993902483109918506715590 1731530135509806316882375 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 192 2228479245828732448682130 1935239563216842354208915 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 193 2697632771266360332615210 2342658418630914428861995 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 194 3401363059422802158514830 2953786701752022540841615 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 195 3635939822141616100481370 3157496129459058578168155 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 196 4339670110298057926380990 3768624412580166690147775 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 197 4808823635735685810314070 4176043267994238764800855 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 198 5043400398454499752280610 4379752695701274802127395 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 199 5512553923892127636213690 4787171551115346876780475 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 200 6216284212048569462113310 5398299834236454988760095 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 201 6920014500205011288012930 6009428117357563100739715 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 202 7154591262923825229979470 6213137545064599138066255 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 203 7858321551080267055879090 6839699495691596202234803 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 204 15716643102160534111758180 13679398991383192404731749 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 205 23574964653240801167637270 20519098487074788607490839 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 206 39291607755401335279395450 34198497478457981013009019 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 207 55008250857561869391153630 47877896469841173418527199 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 208 86441537061882937614669990 75236694452607558229563559 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 209 102158180164043471726428170 88916093443990750635081739 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 210 133591466368364539949944530 116274891426757135446118099 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 211 149308109470525074061702710 129954290418140327851636279 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 212 180741395674846142285219070 157313088400906712662672639 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 213 227891324981327744620493610 198351285375056289879227179 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 214 243607968083488278732251790 212030684366439482284745359 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 215 290757897389969881067526330 253068881340589059501299899 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 216 322191183594290949291042690 280427679323355444312336259 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 217 337907826696451483402800870 294107078314738636717854439 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 218 369341112900772551626317230 321465876297505021528890799 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 219 416491042207254153961591770 362504073271654598745445339 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 220 463640971513735756296866310 403542270245804175961999879 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 221 479357614615896290408624490 417221669237187368367518059 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 222 526507543922377892743899030 458259866211336945584072599 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 ////// Observations ////// 1. The first 27 terms (i.e., the terms m < 2310) include three types of numbers: a. Let k = any product of primorial A002110(i - 1) and the smallest i primes, b. Products k times a prime p >= prime(i) such that k < A002110(i + 1). 2. Let m with MN(m) that contains a zero be called a "turbulent" term, and m with MN(M) consisting of exponents greater than 0 be considered "contiguous". The 16 "turbulent" terms m appear within the first 23 terms and are squarefree except for 126, which is the product of A002110(2), prime(2), and prime(4). 2. Conjectural terms appear above after n = 53 (asterisked). The algorithm h(x) that furnished the conjectured terms tests all numbers of the form described in Observation 1 above. We can easily produce conjectural terms up to primorial A002110(40). ////// Conjectures ////// 1. For m >= A002110(5) = 2310, all terms m are primorials (in A002110) or of the form p * A002110(n), with prime(1) <= prime p <= prime(i). ////// Algorithms ////// A294492: Generate terms m < 10^6 (smallest 41 terms): -------- Block[{s = Array[(# - (DivisorSigma[0, #] + EulerPhi@ # - 1))/# &, 10^6]}, FirstPosition[s, #][[1]] & /@ Union@ FoldList[Max, s]] Necessary-but-insufficient condition h(x): ------------------------------------------ h[n_] := If[n == 0, {1}, Block[{P = Product[Prime@ i, {i, n - 1}], Q = {1}~Join~Prime@ Range@ n, f}, f[p_] := Prime@ Range[n, PrimePi@ NextPrime[Apply[Times, Prime[n + {0, 1}]]/p, -1]]; Union@ Flatten@ Map[P # f[#] &, Q]]] Multiplicity notation: ---------------------- A054841[n_] := If[n == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ n] Produce chart: -------------- Block[{r = Flatten@ Array[h, 20, 0], s, t, u}, s = Map[(# - (DivisorSigma[0, #] + EulerPhi@ # - 1)) &, r]; t = MapIndexed[#1/r[[First@ #2]] &, s]; u = Union@ FoldList[Max, MapIndexed[#1/r[[First@ #2]] &, s]]; MapIndexed[{#1, #2, s[[FirstPosition[r, #2][[1]] ]], StringTrim[StringDelete[ToString@ #, ","], ("{" | "}") ...] &@ A054841@ #2} & @@ {First@ #2, r[[FirstPosition[t, #1][[1]] ]]} &, u]] // TableForm (eof)