Problems & Puzzles: Puzzles

Puzzle 1212 A381732 Paolo Lava sent on March 11, 2025, the following interesting puzzle: Please have a look to A381732. I (Paolo) computed the first 18 terms and Michael Branicky extended their number up to 29. So far the only found prime is 11819327599. These 29 terms are as follow: 27, 737, 909, 1845, 1912, 7078, 27412, 90009, 870129, 990099, 6852899, 9090909, 17388261, 70168376, 70787078, 96096078, 96707298, 162533711, 358006673, 737737737, 1050889491, 2238028254, 3281718034, 4249370147, 9009009009, 11819327599, 12178217823, 13851266943, 18768863945 Examples: 1) 27 is a term since between 2 and 7 we have 3456 and 3456 / 27 = 128 2) 1845 is a term since between 1 and 8 we have 234567, between 8 and 4 765 and between 4 and 5 no digit to be added and 234567765 / 1845 = 127137 Q1. Extend this sequence as far as you can. Q2. Are there other primes in this sequence?

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Contributions came from Giorgos Kalogeropoulos, Simon Cavegn, Emmanuel Vantieghem, Oscar Volpatti

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Giorgos wrote:

Q1. Next terms are 90009090009 and 153660008951

Here are some other composite terms of this sequence that were made by concatenating integers like

 {737, 7078, 870129, 8027289, 17388261, 96707298, 162533711...} or palindromes using digits 0 and 9: 

80272898027289

737737737737737

909000000000909

1738826117388261

7078707870787078

9670729896707298

870129870129870129

70787078707870787078

999000999000999000999

162533711162533711162533711

we can produce infinitely many terms like these

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Simon wrote:

Q1 Next 4 numbers in the sequence:
90009090009
153660008951
320987652667
632583352849

Additional solutions: For example 181181181181181181181181181181181181181181181 ("181" repeated 15 times)
181 reps:15,30,45,...,945,975,990
373 reps:31,93,155,217,279,341,403,465,527,589,651,713,775,837,899,961
707 reps:101,303,505,707,909
737 reps:1,3,5,7,9,11,13...991,993,995,997,999
1812 reps:25,50,100,125,175,200,250,275,325,350,400,425,475,500,550,575,625,650,700,725,775,800,850,875,925,950
2083 reps:347,694
2181 reps:121,242,484,605,847,968
2571 reps:107,321,535,749,963
3833 reps:479
4793 reps:599
4893 reps:29,87,145,203,261,319,377,435,493,551,609,667,725,783,841,899,957
6167 reps:55,165,275,385,495,605,715,825,935
6217 reps:777
7007 reps:7,14,28,...973,980,994
7078 reps:1,2,4...,995,997,998
7077 reps:28,56,112,140,196,224,280,308,364,392,448,476,532,560,616,644,700,728,784,812,868,896,952,980
7227 reps:73,219,365,511,657,803,949
7398 reps:137,411,685,959
7707 reps:61,122,244,305,427,488,610,671,793,854,976
8017 reps:668
8129 reps:41,82,164,205,287,328,410,451,533,574,656,697,779,820,902,943
8369 reps:523
8917 reps:185,370,740,925
10171 reps:363
12481 reps:891
13861 reps:83,249,415,581,747,913
14041 reps:369
14371 reps:171,513,855
14851 reps:99,297,495,693,891
15191 reps:69,207,345,483,621,759,897
15851 reps:143,429,715
16291 reps:37,111,185,259,333,407,481,555,629,703,777,851,925,999
16841 reps:153,459,765
17171 reps:111,333,555,777,999
17281 reps:157,471,785
17401 reps:435
17851 reps:357
18161 reps:21,63,105,147,189,231,273,315,357,399,441,483,525,567,609,651,693,735,777,819,861,903,945,987
19151 reps:87,261,435,609,783,957
19481 reps:33,99,165,231,297,363,429,495,561,627,693,759,825,891,957
20923 reps:28,56,112,140,196,224,280,308,364,392,448,476,532,560,616,644,700,728,784,812,868,896,952,980
20971 reps:104,208,416,520,728,832
22171 reps:739
22481 reps:281,843
22701 reps:759
22841 reps:15,45,75,105,135,165,195,225,255,285,315,345,375,405,435,465,495,525,555,585,615,645,675,705,735,765,795,825,855,885,915,945,975
23933 reps:393
24193 reps:124,248,496,620,868,992
24493 reps:159,477,795
26161 reps:872
26681 reps:667
27493 reps:482,964
28483 reps:104,208,416,520,728,832
28771 reps:137,411,685,959
28903 reps:688
29143 reps:320,640
29693 reps:86,172,344,430,602,688,860,946
29773 reps:783
30173 reps:3,9,15,21,...987,993,999
30513 reps:363
30903 reps:515
36043 reps:9,27,45,...,981,999
37193 reps:429
37583 reps:87,261,435,609,783,957
39893 reps:69,207,345,483,621,759,897
39934 reps:243,729
40063 reps:607
40393 reps:434,868
40634 reps:923
41173 reps:441
41734 reps:3,9,15,...,987,993,999
42913 reps:220,440,880
43903 reps:476,952
43934 reps:499
45193 reps:70,140,280,350,490,560,700,770,910,980
48065 reps:89,178,356,445,623,712,890,979
48133 reps:189,567,945
50645 reps:723
50834 reps:242,484,968
59054 reps:518
60065 reps:73,219,365,511,657,803,949
60665 reps:551
60845 reps:329,658
61007 reps:517
63076 reps:303,909
63077 reps:901
63956 reps:29,87,145,203,261,319,377,435,493,551,609,667,725,783,841,899,957
65077 reps:551
66937 reps:9,27,45...,981,999
69187 reps:469,938
69307 reps:4,8,16,...,980,988,992
70367 reps:429
70837 reps:681
77176 reps:219,657
78127 reps:93,279,465,651,837
78397 reps:509
80697 reps:242,484,968
81417 reps:646
82498 reps:747
84097 reps:308,616
84619 reps:254,508
87298 reps:341
88517 reps:309,927
90919 reps:319
94289 reps:111,333,555,777,999
95969 reps:45,135,225,315,405,495,585,675,765,855,945
98249 reps:99,297,495,693,891
98329 reps:319,957
99309 reps:394,788

Palindromes, only using 0 and 9:
909 reps:1
90009 reps:1
990099 reps:1
9090909 reps:1
9009009009 reps:1
90009090009 reps:1
909000000000909 reps:1
9009000000009009 reps:1,2,4,...994,995,997,998
9000009000009000009 reps:1
99000099000099000099 reps:1
999000999000999000999 reps:1
90009009000900900090090009 reps:1
990000990000990099990099000099000099 reps:1
90090090090090099099099009009009009009 reps:1
90900009090000000000000000009090000909 reps:1
900090090009009090909909090900900090090009 reps:1
900900909909009099090090990900909909009009 reps:1
90900909909099909099999099999090999090990900909 reps:1
900909900990900990090990099090099009099009909009 reps:1,2,3...996,997,998,999
90000009000000900000090000009000000900000090000009 reps:1
90990900900900909909000000000090990900900900909909 reps:1
900990999099009000000000000000000000900990999099009 reps:1
9000090000900009900099000990009900099000090000900009 reps:1
9000090090909900999099909990999099909990099090900900009 reps:1
9000900900090090909099090909009090909909090900900090090009 reps:1
9009009099090090990900909909009099090090990900909909009009 reps:1
9009009009009099099090090090990990900900909909909009009009009 reps:1
900099000990009090009900999099909990999009900090900099000990009 reps:1
909090090909909090090909000000000000000000909090090909909090090909 reps:1

Palindromes, only using 0 and 7:
707007077070777070770700707 reps:1
7007700770077007700770077007 reps:1,2,4,...,995,997,998
70000700707077007770077070700700007 reps:1
77000770777707707777077077770770777707700077 reps:1


Q2
There are probably more primes in the sequence, however I could not find any

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Emmanuel wrote:

I could not find the next term of the sequence  A381732 but I could find many greater terms :
90009090009, 909000000000909, 9009000000009009, 9000009000009000009, 99000099000099000099,
999000999000999000999, 90009009000900900090090009
I conjecture that there are infinitely many numbers with only digits 0, 9  in the sequence.
I also found that  2^n  repetitions of  7078, 17388261, 70787078, 96707298, 1050889491  are in the sequence.
All  3^n repetition of  737, 870129, 162533711  and  358006673.are seemingly in the sequence.
I'm convinced that there are easy proofs for these statements (but I need more time to write them down).

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Oscar wrote:

I performed an exaustive search up to 10^12.

Next four terms of OEIS sequence A381732 are composite: 

90009090009

153660008951

320987652667

632583352849

Then I tried to detect some larger terms of the sequence.

Let x be the starting number and y=f(x) be the resulting number,

where x has dx digits and y has dy digits.

I searched for solutions (x,y) with dx=13 and dy<=30, finding one more term:

2855133980176

Maybe not the 34th term; unfortunately, another composite.

Five terms only contain digits 0 and 9: 

909, 90009, 9090909, 9009009009, 90009090009

I searched for more terms of this form, up to dx = 74:

909000000000909

9009000000009009

9000009000009000009

99000099000099000099

999000999000999000999

90009009000900900090090009

90090000000090099009000000009009

990000990000990099990099000099000099

90090090090090099099099009009009009009

90900009090000000000000000009090000909

900090090009009090909909090900900090090009

900900909909009099090090990900909909009009

90900909909099909099999099999090999090990900909

900909900990900990090990099090099009099009909009

90000009000000900000090000009000000900000090000009

90990900900900909909000000000090990900900900909909

900990999099009000000000000000000000900990999099009

9000090000900009900099000990009900099000090000900009

9000090090909900999099909990999099909990099090900900009

9000900900090090909099090909009090909909090900900090090009

9009009099090090990900909909009099090090990900909909009009

9009009009009099099090090090990990900900909909909009009009009

900099000990009090009900999099909990999009900090900099000990009

9009000000009009900900000000900990090000000090099009000000009009

909090090909909090090909000000000000000000909090090909909090090909

99000099000099009999009900009900999900990000990099990099000099000099

900900090090909909090090009009000000000000900900090090909909090090009009

90009009000900909090990909090090909099090909009090909909090900900090090009

As noted on OEIS sequence page for terms 737 and 7078, the concatenation of term x with itself, repeated r times (r*dx digits) may generate new terms of the sequence.

I detected nine more such infinite families.

dy=2*dx, every r coprime with 2

737, 870129, 162533711, 358006673, 153660008951

dy=3*dx, every r coprime with 3 

7078, 17388261, 96707298, 1050889491, 9009000000009009

dy=5*dx, every r coprime with 5

900909900990900990090990099090099009099009909009

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