Problems & Puzzles: Puzzles

Puzzle 178. Shallit Minimal Primes Set

Believe it or not the following set S of 26 primes:

S = {2,3,5,7,11,19,41,61,89,409,449,499,881,991,6469,6949,9001,9049, 9649,9949,60649,666649,946669,60000049,66000049,66600049}*

has the following very nice & interesting property:

Any prime or is already a member of S or it can be transformed to at least one of these 26 primes in S, by the simple and only procedure of eliminating one or several digits.

This was formally and completely proved by Jeffrey Shallit. See the proof here. See also this page.

Example: the prime 1031 can be transformed to the prime 3 (eliminating the digits 1,0 & 1) or to the prime 11 (eliminating the digits 0 & 3).

Example: The prime 911 can not be transformed to the prime 19, just to the prime 11.

1. Find the minimal prime P_S that can be transformed to each one of the first K primes - one at the time - of the minimal set S obtained by Shallit.

Example:

The prime 12341579 is the minimal prime that can be reduced to each of the first seven (K=7) primes in S (See below the table of the first results)

2. Find the corresponding minimal set of primes S if the universe of primes is restricted to the following types of primes:

a) 4k+1
b) 4k-1
c) palindromic primes.

_____________
* While Shallit obtained that set by a complete logical analysis of the string possibilities of the universe of the primes, we can produce them by a recursive procedure - that for sure you'll imagine very soon. But using this last approach you can never know when you have finished the generation of the minimal set of primes.

As a matter of fact, I asked to Shallit for a halt rule to the recursive procedure. Two minutes later he answered:

Solution:

Table of results for question 1:

***

Andrew Rupinski also got the solution for K=26:

***

Andrew Rupinsik wrote the following solution for question 4a., on 18/4/06. His own comments are:

I have what I believe to be complete solutions to problems 2a and 2b of puzzle 178. The requisite lists of primes are attached.

 

The 173 primes in the set of minimal 4k+1 primes range from the 1 digit prime 5 to the 633 digit (probable) prime 10^633-507.

 

In the 4k-1 case, the range of the 138 members is even wider: from the 1 digit prime 3 to the 19153 digit (probable) prime 2*(10^19153-1)/9+77.

 

When I first began computing, it seemed like it would be a daunting task to compute these sets, but as I went along, I got a lot of small primes which helped reduce the number of calculations of larger primes. Overall I probably tested around 2000-3000 numbers (with as many as 180 cases for particular digit combinations), then compared the resulting primes to weed out the minimal members. I believe all the listed primes are pairwise incomparable in the two lists, although I may have missed a few so these lists might have one or two elements that shouldn't still be there. Conversely, I think that I considered all the cases accurately, although in some digit combinations there were so many numbers to check that I might have overlooked a small subset. Overall, these lists ought to be 99.9% accurate. If anyone is crazy enough to want to check the work, I have left the numbers roughly in the order I considered them so someone could theoretically follow the cases through in the order I did.

 

After doing all this work, I realized that I could have made the work a lot easier if I had worked in base 2 in which case the 4k+1 list consists of just 5 and the 4k-1 list consists of just 3 ;)

On December 27, 2020, Richard Chen wrote:

The minimal set of primes in base 10 is {2,3,5,7,11,19,41,61,89,409,449,499,881,991,6469,6949,9001,9049, 9649,
9949,60649,666649,946669,60000049,66000049,66600049}, and it can be generalized to other bases.

 

Now, I found the minimal set of primes with at least two digits in base 10 and some other bases:

 

Base 10:

 

{11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449,
 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801,
 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649,
 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}

 

Base 2:

 

{10, 11}

 

Base 3:

 

{10, 12, 21, 111}

 

Base 4:

 

{11, 13, 23, 31, 221}

 

Base 5:

 

{10, 12, 21, 23, 32, 34, 43, 111, 131, 133, 313, 401, 414, 14444, 30301, 33001, 33331, 44441, 300031}

 

Base 6:

 

{11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}

 

Base 7:

 

{10, 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346,
 364, 445, 515, 533, 535, 544, 551, 553, 1112, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504,
 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 30011, 31111, 33001, 33311, 35555, 40054, 300053,
 33333301, 33333333333333331}

 

Base 8:

 

{13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301,
 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331,
 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411,
 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441,
 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444
44444444444444444444444444444444444444444444444444444444444444444444444444444444444444
4444444444444444444444444444444444444444444444447}

 

The largest prime in base 8 can be written as 4₂₂₀7 and equal the prime (2^665+17)/7

 

Also, I found the set for base 12:

 

{11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}

 

***

On June 4, 2021, Richard Chen added:

The minimal set of the primes are:

 

* Primes of the form 4k+1: {5, 13, 17, 29, 37, 41, 61, 73, 89, 97, 101, 109, 149, 181, 233, 277, 281, 349, 409, 433, 449, 677, 701, 709, 769, 821, 877, 881, 1669, 2221, 3001, 3121, 3169, 3221, 3301, 3833, 4969, 4993, 6469, 6833, 6949, 7121, 7477, 7949, 9001, 9049, 9221, 9649, 9833, 9901, 9949, 11969, 19121, 20021, 20201, 21121, 23021, 23201, 43669, 44777, 47777, 60493, 60649, 66749, 80833, 90121, 91121, 91921, 91969, 94693, 111121, 112121, 119921, 199921, 220301, 466369, 470077, 666493, 666649, 772721, 777221, 777781, 779981, 799921, 800333, 803333, 806033, 833033, 833633, 860333, 863633, 901169, 946369, 946669, 999169, 1111169, 1999969, 4007077, 4044077, 4400477, 4666693, 8000033, 8000633, 8006633, 8600633, 8660033, 8830033, 8863333, 8866633, 22000001, 40400077, 44040077, 60000049, 66000049, 66600049, 79999981, 80666633, 83333333, 86606633, 86666633, 88600033, 88883033, 88886033, 400000477, 400444477, 444000077, 444044477, 836666333, 866663333, 888803633, 888806333, 888880633, 888886333, 8888800033, 8888888033, 88888883333, 440444444477, 7777777777921, 8888888888333, 40000000000777, 44444444400077, 40444444444444477, 44444444444444477, 88888888888888633, 999999999999999121, 8888888888888888888888888888888888888888888888888888888888888888888888888888833}

 

* Primes of the form 4k+3: {3, 7, 11, 19, 59, 251, 491, 499, 691, 991, 2099, 2699, 2999, 4051, 4451, 4651, 5051, 5651, 5851, 6299, 6451, 6551, 6899, 8291, 8699, 8951, 8999, 9551, 9851, 22091, 22291, 66851, 80051, 80651, 84551, 85451, 86851, 88651, 92899, 98299, 98899, 200891, 208891, 228299, 282299, 545551, 608851, 686051, 822299, 828899, 848851, 866051, 880091, 885551, 888091, 888451, 902299, 909299, 909899, 2000291, 2888299, 2888891, 8000099, 8000891, 8000899, 8028299, 8808299, 8808551, 8880551, 8888851, 9000451, 9000899, 9908099, 9980099, 9990899, 9998099, 9999299, 60000851, 60008651, 60086651, 60866651, 68666651, 80088299, 80555551, 80888299, 88808099, 88808899, 88880899, 90000299, 90080099, 222222899, 800888899, 808802899, 808880099, 808888099, 888800299, 888822899, 992222299, 2222288899, 8808888899, 8888800099, 8888888299, 8888888891, 48555555551, 555555555551, 999999999899, 88888888888099, 2228888888888899, 9222222222222299, 2288888888888888888888899, 888888888888888888888888888888888888888888899, 86666666666666666666666666666666666666666666666651, (2_19151)99}

 

* Non-single digit primes: {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}

 

* Palindromic primes: {2, 3, 5, 7, 11, 919, 94049, 94649, 94849, 94949, 96469, 98689, 9809089, 9888889, 9889889, 9908099, 9980899, 9989899, 900808009, 906686609, 906989609, 908000809, 908444809, 908808809, 909848909, 960898069, 968999869, 988000889, 989040989, 996686699, 996989699, 999686999, 90689098609, 90899999809, 90999899909, 96099899069, 96600800669, 96609890669, 98000000089, 98844444889, 9009004009009, 9099094909909, 9600098900069, 9668000008669, 9699998999969, 9844444444489, 9899900099989, 9900004000099, 9900994990099, 900006898600009, 900904444409009, 966666989666669, 966668909866669, 966699989996669, 999090040090999, 999904444409999, 90000006860000009, 90000008480000009, 90000089998000009, 90999444444499909, 96000060806000069, 99900944444900999, 99990009490009999, 99999884448899999, 9000090994990900009, 9000094444444900009, 9666666080806666669, 9666666668666666669, 9909999994999999099, 9999444444444449999, 9999909994999099999, 9999990994990999999, 900000000080000000009, 900999994444499999009, 90000000009490000000009, 90909444444444444490909, 98999999444444499999989, 9904444444444444444444099, 999999999844444448999999999, 90944444444444444444444444909, 99999999999944444999999999999, 99999999999999499999999999999, 9999999999990004000999999999999, 900000000999999949999999000000009, 989999999999998444899999999999989, 9000000999999999994999999999990000009, 99(4_34019)99}

 

* Non-palindromic primes: {13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 211, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 811, 821, 827, 857, 877, 881, 887, 911, 991, 1021, 1051, 1151, 1181, 1201, 1511, 1801, 2087, 2221, 5011, 5051, 5081, 5101, 5501, 5581, 5801, 5851, 6469, 6949, 7027, 7057, 7207, 7507, 7727, 7757, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 10111, 20021, 20201, 50207, 51551, 55511, 60649, 78007, 80051, 111121, 150001, 185551, 511111, 666649, 700087, 777787, 946669, 1000081, 1100101, 5200007, 7700807, 11111101, 22000001, 60000049, 66000049, 66600049, 80555551, 11100000001, 11110000001, 555555555551, 5000000000000000000000000000027}

 

* Primes of the form 3k+1: {7, 13, 19, 31, 43, 61, 151, 181, 211, 223, 229, 241, 283, 349, 409, 421, 499, 523, 541, 811, 823, 829, 853, 859, 883, 991, 1021, 1201, 2053, 2089, 2221, 2251, 2281, 2389, 2503, 2521, 2539, 2551, 2593, 2659, 2689, 2851, 2953, 3253, 3259, 3529, 3559, 3583, 3889, 4051, 4111, 4441, 4549, 4591, 4801, 4951, 5011, 5059, 5101, 5209, 5281, 5449, 5503, 5521, 5563, 5569, 5581, 5653, 5659, 5683, 5689, 5821, 5839, 5851, 5869, 5881, 5953, 6469, 6529, 6553, 6949, 8089, 8221, 8389, 8521, 8581, 8689, 8821, 8941, 9001, 9049, 9511, 9649, 9949, 10111, 10141, 14011, 14401, 20359, 20509, 20599, 20959, 23059, 23509, 23563, 23599, 24469, 24889, 25609, 25633, 25801, 25849, 25969, 25981, 25999, 26449, 28069, 28081, 28099, 28201, 28669, 28909, 28921, 29059, 29569, 29581, 29599, 29881, 29959, 29989, 30553, 32869, 33289, 33589, 35053, 35089, 35353, 35533, 35809, 35899, 42589, 42649, 44101, 44269, 44449, 44851, 45259, 45289, 45589, 46489, 48481, 48649, 48889, 49081, 49411, 50053, 50221, 50329, 50383, 50551, 50833, 50893, 50929, 50989, 52021, 52051, 52201, 52489, 52501, 52951, 52999, 53089, 53269, 53299, 53353, 53359, 53593, 53629, 53899, 53959, 54559, 55009, 55051, 55249, 55333, 55339, 55399, 55501, 55849, 55933, 56269, 56299, 56629, 56929, 58099, 58363, 58603, 58909, 58963, 59029, 59083, 59221, 59359, 59509, 59629, 59809, 59833, 59863, 59929, 60259, 60289, 60589, 60649, 60889, 62533, 62563, 62653, 62869, 62989, 63589, 64489, 64849, 65089, 65599, 65809, 65899, 65983, 66889, 68449, 68899, 69259, 80251, 80449, 80491, 82051, 84481, 84649, 88069, 88609, 88801, 88951, 88969, 89449, 89809, 89899, 89989, 90121, 90289, 90481, 90529, 90583, 90841, 91141, 91411, 92353, 92569, 92581, 92809, 92821, 92899, 92959, 93553, 94849, 95083, 95089, 95383, 95539, 95629, 95803, 95929, 95959, 95989, 96259, 96289, 96589, 98041, 98251, 98809, 98869, 98899, 99259, 99289, 99529, 99559, 100411, 111121, 114001, 200569, 200881, 205081, 205549, 209449, 209821, 233353, 236653, 255049, 255589, 256699, 259009, 259459, 263533, 265333, 266353, 280009, 286009, 286999, 288049, 288649, 289489, 289999, 290821, 295459, 298021, 298999, 300589, 302989, 306589, 309289, 329899, 353389, 359389, 404011, 408841, 410401, 424849, 442489, 444289, 444529, 444589, 446569, 449011, 450001, 484489, 491041, 500029, 500083, 500299, 500389, 500629, 500809, 502669, 502699, 503389, 504289, 508009, 508489, 511111, 525949, 530533, 530983, 533389, 533809, 533893, 533989, 535099, 535999, 538093, 538303, 538333, 539389, 550489, 552589, 555589, 559099, 559459, 559549, 559939, 560029, 580033, 580093, 580303, 580633, 580663, 586633, 588949, 589903, 589933, 589993, 590251, 590389, 590899, 592849, 594889, 595549, 598489, 598903, 598999, 599383, 599803, 599899, 599959, 599983, 604249, 605509, 606559, 609253, 620569, 623353, 625489, 625699, 625909, 629509, 645889, 650863, 653083, 653893, 654889, 655489, 658303, 658633, 658663, 660559, 662059, 662353, 662449, 662899, 664459, 665359, 665803, 666559, 666649, 669289, 684889, 688669, 689869, 695389, 695509, 696253, 699253, 800281, 805501, 808081, 808441, 825001, 840841, 844489, 848851, 880909, 880981, 884491, 884881, 886999, 888451, 888469, 889081, 889489, 889909, 890551, 890881, 895051, 898669, 900253, 900259, 900553, 900589, 901111, 901441, 902563, 902599, 902653, 905053, 905551, 905599, 908851, 908881, 909253, 909889, 911011, 911101, 914041, 925663, 926533, 928849, 929869, 941041, 942889, 944551, 944659, 944689, 946669, 950029, 950251, 950269, 952669, 953053, 954259, 955993, 958333, 958849, 958933, 966583, 969253, 969889, 980851, 982801, 988051, 988489, 988501, 988849, 990589, 990889, 993589, 995053, 995833, 995983, 996253, 998989, 999553, 1100041, 1100101, 2004559, 2005459, 2080021, 2500009, 2500081, 2500669, 2535553, 2536663, 2555353, 2588809, 2588899, 2595559, 2656663, 2665363, 2880901, 2888449, 2950009, 2955559, 2955889, 3000289, 3002899, 3653989, 3662809, 3665989, 3909589, 4000081, 4004881, 4014001, 4140001, 4446259, 4455001, 4455559, 4484869, 4642459, 4644259, 4884469, 5000251, 5008063, 5009803, 5028949, 5080003, 5255959, 5266969, 5295559, 5298889, 5300803, 5300863, 5308663, 5330803, 5333329, 5333929, 5338033, 5338633, 5380003, 5380009, 5389003, 5393329, 5528899, 5528989, 5529889, 5551111, 5555509, 5555929, 5555983, 5559259, 5559529, 5559889, 5595559, 5598889, 5599999, 5830333, 5888899, 5889889, 5938003, 5939893, 5952559, 5952889, 5955259, 5955529, 5958889, 5995999, 6004429, 6004459, 6005383, 6005863, 6020449, 6058033, 6058333, 6059959, 6060583, 6066583, 6090559, 6095359, 6095833, 6200059, 6233959, 6250009, 6250099, 6256669, 6288889, 6293359, 6366289, 6442459, 6444259, 6508003, 6533383, 6533803, 6533833, 6539989, 6599389, 6623959, 6624589, 6625459, 6625669, 6653389, 6653833, 6653989, 6655549, 6662809, 6662959, 6663289, 6664429, 6665383, 6666589, 6695863, 6880009, 6900559, 6999589, 8000401, 8005051, 8020801, 8048881, 8090851, 8092801, 8208001, 8288881, 8488981, 8505001, 8550001, 8800009, 8800501, 8800999, 8808841, 8840401, 8844001, 8850001, 8884849, 8885551, 8886649, 8888449, 8888491, 8888851, 8888989, 8889889, 9005509, 9025333, 9058999, 9060559, 9095899, 9099589, 9250009, 9250909, 9253333, 9266563, 9284449, 9424669, 9444889, 9525589, 9550999, 9558889, 9559909, 9559999, 9600559, 9605359, 9606253, 9844501, 9844669, 9845551, 9884551, 9888481, 9888841, 9888889, 9926563, 9953389, 9953533, 9953809, 9958999, 9965863, 9992533, 9996583, 9999889, ...}  (This set is very large, may not be complete)

 

*  Primes of the form 3k+2: {2, 5, 11, 17, 41, 47, 71, 83, 89, 149, 443, 449, 677, 743, 773, 797, 881, 887, 977, 1433, 3767, 3779, 6143, 7079, 7307, 7349, 7499, 7607, 7649, 7877, 7949, 9749, 10343, 10463, 13043, 13463, 16943, 19403, 19463, 30707, 31643, 33377, 33749, 36749, 37049, 37337, 37409, 66749, 67049, 67409, 70067, 70667, 73379, 73637, 73679, 73709, 74609, 76367, 76379, 76667, 76679, 77069, 77687, 77699, 77867, 77969, 77999, 80777, 87767, 91463, 91943, 100043, 100403, 100943, 104003, 109943, 133403, 136343, 136403, 139343, 139943, 140603, 140663, 146063, 146603, 160403, 163403, 163643, 164663, 166043, 166403, 166643, 193943, 196043, 196643, 199343, 300749, 301403, 301463, 301943, 303143, 310043, 313343, 314003, 314063, 314603, 319343, 331043, 331943, 338777, 374669, 387077, 387707, 391403, 393143, 603749, 674669, 704009, 704069, 706337, 767909, 770909, 900143, 901403, 901643, 903143, 910643, 991043, 991343, 991643, 1339643, 1460003, 1909043, 1933643, 1990643, 1999043, 3000077, 3000377, 3007787, 3199943, 3301343, 3309143, 3331463, 3339143, 3377807, 3390143, 3391343, 3707777, 3770807, 3778007, 3901043, 3999143, 7000337, 7007087, 7033787, 7070087, 7333367, 7337777, 7400069, 7633337, 7676609, 7700807, 7707767, 7708007, 7766009, 7777409, 7777667, 7777709, 7777769, 7780007, 8707007, 8770007, 9136643, 9140003, 9301043, 9333143, 9909143, 9930143, ...}  (This set is very large, may not be complete)

 

* Emirps: {13, 17, 31, 37, 71, 73, 79, 97, 149, 199, 359, 389, 941, 953, 983, 991, 1009, 1021, 1061, 1069, 1091, 1109, 1151, 1181, 1201, 1229, 1259, 1511, 1559, 1601, 1619, 1669, 1811, 1901, 3023, 3049, 3083, 3203, 3299, 3343, 3433, 3463, 3469, 3583, 3643, 3803, 3853, 3929, 7027, 7057, 7207, 7457, 7507, 7547, 7577, 7687, 7757, 7867, 9001, 9011, 9029, 9161, 9209, 9221, 9293, 9349, 9403, 9439, 9521, 9551, 9601, 9643, 9661, 9923, 10159, 10859, 10889, 11159, 11161, 11621, 12119, 12241, 12611, 12641, 12689, 12809, 12841, 14081, 14221, 14251, 14551, 14621, 14821, 15241, 15289, 15461, 15541, 15661, 16111, 16451, 16481, 16651, 16829, 18041, 18089, 18169, 18269, 18461, 18691, 18859, 19681, 30029, 32009, 32233, 32353, 32369, 32563, 32633, 32693, 32933, 33029, 33223, 33329, 33623, 33863, 33923, 35323, 35363, 36209, 36269, 36353, 36523, 36833, 39623, 70667, 74077, 76607, 77047, 90023, 90059, 90089, 90263, 90499, 90821, 90989, 91121, 91129, 92003, 92033, 92119, 92189, 92333, 92369, 92459, 92489, 92639, 92861, 92899, 92959, 93629, 94559, 94889, 95009, 95101, 95111, 95429, 95549, 95801, 95881, 95929, 96181, 96263, 96281, 96289, 96323, 96329, 98009, 98081, 98129, 98251, 98269, 98299, 98429, 98621, 98801, 98849, 98909, 98999, 99289, 99409, 99829, 99989, 100411, 101119, 101141, 104801, 108401, 108881, 111119, 111211, 111829, 111869, 112111, 114001, 114041, 116911, 119611, 121421, 121921, 124121, 125551, 125651, 125821, 126551, 126851, 128521, 128551, 129121, 129281, 140411, 141101, 144481, 144541, 145441, 145861, 146161, 146581, 152981, 155521, 155581, 155621, 155821, 155851, 156521, 158551, 158621, 158981, 161641, 162881, 168541, 182921, 184441, 185291, 185551, 185641, 185681, 185819, 186581, 186889, 188261, 188609, 188801, 189251, 189851, 192581, 302609, 302629, 305603, 305633, 306329, 306503, 322229, 322249, 322649, 323333, 324293, 328883, 329663, 330053, 333253, 333323, 333563, 335633, 335653, 336503, 336533, 338383, 340453, 340909, 340999, 344053, 344293, 344453, 344843, 348443, 349399, 349493, 350033, 350443, 350663, 352333, 354043, 354443, 356533, 362293, 362339, 362449, 363683, 364909, 365333, 366053, 366239, 366293, 366409, 366923, 383683, 383833, 386363, 386383, 388823, 392263, 392423, 392443, 392663, 394943, 394993, 399493, 700067, 704447, 704747, 707767, 724747, 725587, 725827, 727487, 727877, 728527, 728747, 740087, 744407, 746267, 746747, 747407, 747427, 747647, 747827, 748877, 755267, 760007, 762557, 762647, 766477, 767707, 774667, 777787, 777877, 778727, 778777, 778847, 780047, 780887, 784727, 785527, 787777, 788087, 904459, 904663, 906203, 906881, 906949, 908549, 908959, 909043, 909463, 909599, 911101, 911111, 915869, 918581, 918829, 921629, 922223, 922499, 923603, 924299, 926129, 926203, 928111, 928159, 928289, 928469, 928819, 928859, 932663, 933263, 940469, 942223, 942569, 944263, 945089, 945809, 946223, 946459, 946949, 948659, 949609, 949649, 950569, 951089, 951829, 951859, 952859, 954409, 954649, 954869, 956569, 956689, 956849, 958159, 958259, 958829, 959489, 959689, 959809, 959869, 960889, 964049, 964829, 965059, 965189, 965249, 965659, 965969, 968111, 968459, 968519, 968959, 969569, 980159, 980549, 981569, 982829, 984959, 986659, 986959, 988069, 988681, 992429, 993943, 994229, 995699, 995909, 996599, 999043, 1005581, 1040141, 1041041, 1054441, 1055881, 1080851, 1100441, 1111921, 1119121, 1124141, 1126661, 1144441, 1164461, 1164641, 1165889, 1166441, 1185689, 1219111, 1222681, 1226581, 1228651, 1248881, 1252681, 1255861, 1262581, 1268261, 1286881, 1291111, 1298581, 1298651, 1400051, 1401401, 1410401, 1412461, 1414211, 1414261, 1424681, 1440011, 1444411, 1444501, 1446611, 1464461, 1464611, 1466461, 1486561, 1488481, 1500041, 1508851, 1522681, 1568221, 1568921, 1568951, 1580801, 1585261, 1588051, 1588681, 1589561, 1595861, 1598651, 1624141, 1624661, 1625851, 1628621, 1642141, 1644611, 1644641, 1646641, 1656841, 1659851, 1659881, 1662281, 1664261, 1666211, 1685521, 1685951, 1808581, 1822661, 1848841, 1850129, 1852621, 1855001, 1856221, 1858081, 1858921, 1862221, 1862251, 1862521, 1864241, 1868851, 1882891, 1885501, 1886821, 1888189, 1888421, 1888981, 1889561, 1898881, 1982881, 3009443, 3033053, 3033533, 3065533, 3066229, 3099443, 3220669, 3232429, 3232963, 3233663, 3242429, 3249443, 3282283, 3284483, 3305063, 3322069, 3326629, 3326663, 3332363, 3334099, 3334999, 3339949, 3349999, 3353303, 3355603, 3365563, 3369409, 3422429, 3422453, 3424249, 3424283, 3434099, 3440399, 3442409, 3449003, 3449423, 3449903, 3449909, 3452453, 3494009, 3499099, 3499499, 3499999, 3503303, 3522263, 3542243, 3542543, 3600563, 3605033, 3622253, 3626683, 3628663, 3632333, 3632663, 3632963, 3634949, 3634999, 3649999, 3650063, 3655633, 3662363, 3663323, 3663949, 3664249, 3664499, 3666233, 3668263, 3669499, 3692323, 3692363, 3699499, 3822823, 3824243, 3828683, 3838883, 3844823, 3866263, 3868283, 3888383, 3940099, 3944909, 3969649, 7004087, 7004477, 7004677, 7008047, 7008887, 7047787, 7048477, 7067077, 7074677, 7080077, 7224647, 7224667, 7227287, 7228477, 7242727, 7246727, 7256657, 7262467, 7262677, 7264277, 7272427, 7272647, 7272887, 7276427, 7278827, 7282487, 7287277, 7288727, 7400467, 7408007, 7422887, 7426477, 7426667, 7427447, 7427477, 7427887, 7428887, 7444277, 7446277, 7447067, 7447247, 7447267, 7447777, 7462277, 7462727, 7464227, 7474877, 7477667, 7482287, 7522267, 7526557, 7556257, 7562567, 7565567, 7566527, 7607447, 7622257, 7622767, 7627447, 7640047, 7642627, 7647667, 7652657, 7655657, 7664227, 7666247, 7667467, 7667747, 7672267, 7700807, 7707607, 7722647, 7724447, 7724627, 7726447, 7727827, 7744007, 7746247, 7747247, 7748227, 7748407, 7762627, 7764007, 7764707, 7777447, 7784747, 7788787, 7804007, 7822847, 7827227, 7842827, 7848787, 7877407, 7878487, 7878877, 7882247, 7882727, 7887247, 7888007, 7888247, 9000049, 9004469, 9004669, 9004943, 9015689, 9042443, 9046589, 9048509, 9049633, 9058409, 9059969, 9065449, 9065669, 9068069, 9081659, 9084469, 9088589, 9088699, 9090649, 9094469, 9094493, 9094649, 9094669, 9099443, 9099469, 9118589, 9185689, 9210581, 9222569, 9225589, 9225989, 9226549, 9226603, 9226859, 9229699, 9229949, 9242243, 9242323, 9242423, 9244969, 9252589, 9252949, 9256589, 9256969, 9264449, 9266233, 9268559, 9269699, 9284449, 9285569, 9292949, 9298889, 9299669, 9400009, 9404849, 9406549, 9408569, 9424243, 9424663, 9429499, 9429869, 9444629, 9444829, 9445609, 9446659, 9446989, 9452269, 9456049, 9456229, 9460849, 9460909, 9464099, 9464849, 9464909, 9469693, 9480649, 9484049, 9484649, 9492529, 9492929, 9493663, 9494363, 9494689, 9496259, 9498959, 9499229, 9499333, 9506459, 9506969, 9508669, 9509959, 9522269, 9525589, 9526949, 9545099, 9545969, 9546059, 9546599, 9550669, 9552299, 9555569, 9555599, 9555859, 9556699, 9556889, 9556969, 9558629, 9559699, 9560669, 9561809, 9562499, 9566449, 9568859, 9569089, 9581189, 9584089, 9584699, 9585559, 9585889, 9586229, 9588659, 9588869, 9588989, 9598949, 9599059, 9599069, 9599969, 9602233, 9605669, 9606599, 9606899, 9608609, 9608969, 9609959, 9611869, 9622259, 9622549, 9625669, 9626299, 9626699, 9642599, 9644009, 9644809, 9644909, 9649909, 9652229, 9654989, 9655559, 9655829, 9656099, 9658049, 9660223, 9660559, 9660659, 9662399, 9664009, 9664909, 9665069, 9665269, 9665609, 9666689, 9668059, 9669929, 9681169, 9688859, 9689249, 9689689, 9694429, 9694589, 9695299, 9695459, 9696059, 9696529, 9696559, 9698069, 9698089, 9699509, 9699959, 9804859, 9808969, 9809659, 9811859, 9818689, 9818881, 9844699, 9846989, 9852529, 9854899, 9854969, 9855229, 9855259, 9855689, 9856409, 9856529, 9858119, 9858809, 9864949, 9865109, 9865589, 9865811, 9865819, 9866669, 9868189, 9869869, 9885611, 9885859, 9885989, 9886559, 9888929, 9894569, 9895229, 9895889, 9896449, 9896489, 9898859, 9900493, 9904333, 9904343, 9904649, 9905459, 9906569, 9909943, 9922559, 9925969, 9926269, 9929999, 9930443, 9932669, 9942659, 9944663, 9949249, 9949663, 9949943, 9949963, 9952469, 9955559, 9956069, 9956459, 9962999, 9964489, 9964859, 9966269, 9966559, 9968809, 9969229, 9969559, 9969629, 9984589, 9986069, 9992699, 9994333, 9994363, 9999299, 9999433, 9999463, 9999943, 10011101, 10045001, 10054001, 10054481, 10111001, 10141111, 10404451, 10444051, 10508051, ...} (This set is currently not known, and might be extremely difficult to found)

 

Also, the minimal sets of other sets:
 

 

* Composites: {4, 6, 8, 9, 10, 12, 15, 20, 21, 22, 25, 27, 30, 32, 33, 35, 50, 51, 52, 55, 57, 70, 72, 75, 77, 111, 117, 171, 371, 711, 713, 731}

 

* Squares: {1, 4, 9, 25, 36, 576, 676, 7056, 80656, 665856, 2027776, 2802276, 22282727076, 77770707876, 78807087076, 7888885568656, 8782782707776, 72822772707876, 555006880085056, 782280288087076, 827702888070276, 888288787822276, 2282820800707876, 7880082008070276, 80077778877070276, 88778000807227876, 782828878078078276, 872727072820287876, 2707700770820007076, 7078287780880770276, 7808287827720727876, 8008002202002207876, 27282772777702807876, 70880800720008787876, 72887222220777087876, 80028077888770207876, 80880700827207270276, 87078270070088278276, 88002002000028027076, 2882278278888228807876, 8770777780888228887076, 77700027222828822007876, 702087807788807888287876, 788708087882007280808827876, 880070008077808877000002276, 888000227087070707880827076, 888077027227228277087787076, 888588886555505085888555556, 7770000800780088788282227776, 7782727788888878708800870276, 5000060065066660656065066555556, 8070008800822880080708800087876, 80787870808888808272077777227076, 800008088070820870870077778827876, 822822722220080888878078820887876, ...} (This set is currently not known, and might be extremely difficult to found, although it is known that no repdigits (numbers whose all digits are same) are squares)

 

* Cubes: {1, 8, 27, 64, 343, 729, 3375, 4096, 35937, 39304, 46656, 50653, 79507, 97336, 300763, 405224, 456533, 474552, 493039, 636056, 704969, 3307949, 4330747, 5545233, 5639752, 5735339, 6539203, 9663597, 23393656, 23639903, 29503629, 37933056, 40353607, 45499293, 50243409, 54439939, 57066625, 57960603, 70444997, 70957944, 73560059, 76765625, 95443993, 202262003, 236029032, 350402625, 377933067, 379503424, 445943744, 454756609, 537367797, 549353259, 563559976, 567663552, 773620632, 907039232, ...} (This set is currently not known, and might be extremely difficult to found)

 

* Powers of 2: {1, 2, 4, 8, 65536} (Only conjectured)

 

* Powers of 3: {1, 3, 9, 27} (Only conjectured)

 

* Perfect powers (0 and 1 are not counted): {4, 8, 9, 16, 25, 27, 32, 36, 100, 121, 512, 576, 676, 1331, 2601, 3375, 6561, 7056, 7776, 22201, 50653, 62001, 63001, 505521, 657721, 753571, 5000211, 5067001, 5177717, 5755201, 7557001, ...} (May not be complete)

 

* Powerful numbers (0 and 1 are not counted) : {4, 8, 9, 16, 25, 27, 32, 36, 72, 100, 121, 200, 500, 512, 576, 675, 676, 1152, 1331, 2601, 3375, 6561, 7056, 7776, 15552, 22201, 50653, 60552, 61731, 63001, 77175, 202612, 357075, 505521, 570375, 665523, 735075, 753571, 766656, 1037575, 3501153, 5177717, 6171373, 7555707, 15135133, 15150375, 15151531, ...}  (May not be complete)

 

* Prime powers (1 is not counted): {2, 3, 4, 5, 7, 8, 9, 11, 16, 61}

 

* Squarefree numbers: {1, 2, 3, 5, 6, 7, 89, 94, 409, 449, 498, 499, 998}

***