Problems & Puzzles: Puzzles

Puzzle 1039. k-number of prime divisors obtained by substitution of... Metin Sariyar sent the following nice puzzle:   1080787909 is the smallest prime without digits k={2,3,4,5,6} that contains only  one 1 such that after replacing 1 with any k (2,3,4,5,6) the new number   has k number of prime divisors (multiplicity).  For ex: 61 is prime, 62=2*31 (2 prime divisors) , 63=3*3*7 (3 prime divisors)  1080787909 is prime and  2080787909 = 409 × 5087501,3080787909 = 3 × 71 × 14463793, 4080787909 = 11 × 19 × 757 × 25793,  5080787909 = 172 × 79 × 139 × 1601, 6080787909 = 33 × 7 × 23 × 1398847   Sequence of such numbers I found is: 61(up to 2 ), 61(up to 3), 661(u to 4), 679919 (..5), 1080787909 (..6)    Q. Can you find the next terms  smallest prime without digits k={2,3,4,5,6,7} ... {2,3,4,5,6,7,8}... (with multiplicity or distinct prime divisors and squarefree versions )

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During the week 16-21 May, 2021, contributions came from Giorgos Kalogeropoulos, Oscar Volpatti, Paul Cleary, Simon Cavegn and Emmanuel Vanieghem.

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Giorgos wrote:

number of prime factors counting multiplicities: 
 

1898909980988089 -> k={2,3,4,5,6,7}
190900999999990999909999099099 -> k={2,3,4,5,6,7,8}

1009990090000999000000090009009 ->  k={2,3,4,5,6,7,8}
 

number of distinct prime factors:    

1070077878007 -> k={2,3,4,5,6}

1099089998900989 -> k={2,3,4,5,6,7}

190900999999990999909999099099 -> k={2,3,4,5,6,7,8}

1009990090000999000000090009009 ->  k={2,3,4,5,6,7,8}

number of prime factors in Squarefree numbers:

1098809078077 -> k={2,3,4,5,6}

199988888899880900899 -> k={2,3,4,5,6,7}

190900999999990999909999099099 -> k={2,3,4,5,6,7,8}

1009990090000999000000090009009 ->  k={2,3,4,5,6,7,8}

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Oscar wrote:

Multiplicity version:

k  n

2  61

3  61

4  661

5  679919

6  1080787909
7  88088900999819
8  > 9*10^20


88088900999819 is prime
88088900999829 = 3*29362966999943
88088900999839 = 173*225611*2256913
88088900999849 = 47*919*3391*601423
88088900999859 = 3*3*19*85313*6038233
88088900999869 = 7*11*181*739*2011*4253
88088900999879 = 13*13*31*157*223*283*1697


Squarefree version:

k  n

2  61
3  199
4  167077
5  6907619
6  700709801797
7  > 8*10^17
8  > 9*10^20

 

199 is prime

299 = 13*23

399 = 3*7*19

167077 is prime
 

267077 = 409*653

367077 = 3*37*3307

467077 = 13*19*31*61

6907619 is prime

6907629 = 3*2302543

6907639 = 41*331*509 

6907649 = 7*43*53*433

6907659 = 3* 11*19*23*479

700709801797 is prime

700709802797 = 29*24162406993

700709803797 = 3*313*746229823

700709804797 = 47*337*401*110323

700709805797 = 19*43*199*991*4349

700709806797 = 3*7*11*13*8681*26879

 

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Paul wrote:

I found for k={2,3,4,5,6,7} the following

 

88088900999819 -> {88088900999829, 88088900999839, 88088900999849, 88088900999859,

88088900999869, 88088900999879}

 

with the following factors

 

{{3,1},{29362966999943,1}}

{{173,1},{225611,1},{2256913,1}}

{{47,1},{919,1},{3391,1},{601423,1}}

{{3,2},{19,1},{85313,1},{6038233,1}}

{{7,1},{11,1},{181,1},{739,1},{2011,1},{4253,1}}

{{13,2},{31,1},{157,1},{223,1},{283,1},{1697,1}}

 

For k={2,3,4,5,6,7,8} was not able to solve this but this was my closest

 

10009099099909099090999909 -> {20009099099909099090999909, 30009099099909099090999909,

40009099099909099090999909, 50009099099909099090999909,

60009099099909099090999909, 70009099099909099090999909,

80009099099909099090999909}

 

with the following factors

 

{{570841,1},{35051965608477840749,1}}

{{3,1},{5630843297,1},{1776471570187799,1}}

{{59,1},{27653,1},{985451,1},{24884531716217,1}}

{{61,2},{36083,1},{229685263,1},{1621636801,1}}

{{3,1},{7,1},{13499,1},{525101,1},{19136479,1},{21066449,1}}

{{13,1},{17,1},{23,1},{659,1},{9833,1},{13063,1},{162712155643,1}}

{{37,1},{43,1},{97,1},{113,1},{359,1},{379441,1},{33680638661,1}}

 

The last number has only 7 factors not 8.

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Simon wrote:

Found a solution but its not square free : 88088900999819

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Emmanuel wrote:

The next term is the sequence is  880889000019.  Indeed, we have :
    88 088900 999819 is prime
    88 088900 999829 = 3*29362966999943
    88 088900 999839 = 173*225611*2256913
    88 088900 999849 = 47*919*3391*601423
    88 088900 999859 = 3*3*19*85313*6 038233
    88 088900 999869 = 7*11*181*739*2011*4253
    88 088900 999879 = 13*13*31*157*223*283*1697

 

I spend the rest of the week in searching the next term (which contains only  0s and  9s and only one  1 ). It  must have at least  26 digits.Tomorrow morning I will know if at least 27 digits will be needed

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Giorgos wrote on May 28, 2021:

Minimal number where k is the  number of prime factors counting multiplicities k={2,3,4,5,6,7,8} is  

19009909099090099909909909009  {29 digits}

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