Problems & Puzzles: Puzzles

Puzzle 1153 a new sequence... On Oct 17, Giorgos Kalogeropoulos wrote: Here is the lexicographically earliest sequence of distinct primes with the property that the concatenation of the first n terms produces an integer with n distinct primes: 2, 13, 5, 37, 643, 263, 17159, 337, 6661, 174197, 71551, 901247... n | a(n) | prime factorization of concatenation a(1)|a(2)...a(n) 1 | 2 | 2 = 2 2 | 13 | 213 = 3 * 71 3 | 5 | 2135 = 5 * 7 * 61 4 | 37 | 213537 = 3 * 17 * 53 * 79 5 | 643 | 213537643 = 11 * 29 * 67 * 97 * 103 6 | 263 | 213537643263 = 3^2 * 7^2 * 17 * 31^2 * 107 * 277 7 | 17159 | 21353764326317159 = 11 * 19 * 23 * 83 * 1567 * 3433 * 9949 8 | 337 | 21353764326317159337 = 3^2 * 29 * 31 * 41 * 71 * 97 * 373 * 25058177 9 | 6661 | 213537643263171593376661 = 7 * 19 * 47 * 53 * 701 * 1423 * 1709 * 8171 * 46271 10 | 174197 | 213537643263171593376661174197 = 3 * 31 * 353 * 593 * 1409 * 1669 * 3137 * 7901 * 8447 * 22279 11 | 71551 | 21353764326317159337666117419771551 = 7 * 29 * 61 * 67 * 79 * 257 * 2099 * 13451 * 41513 * 419999 * 2575219 12 | 901247 | 21353764326317159337666117419771551901247 = 3^2 * 13 * 19 * 131 * 167 * 349 * 593 * 9413 * 19559 * 29581 * 55633 * 7002402311 Q. Can you extend this sequence?

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During the week from 2-8 DEc 2023, contributions came from Michael Branicky, JM Rebert, Emmanuel Vantieghem

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Michael wrote:

I confirm the first 12 terms and find the 13th term is 21179423:

 

13 | 21179423 |

2135376432631715933766611741977155190124721179423 =

7^2 * 11 * 23 * 29 * 59 * 229 * 359 * 3299 * 6311 * 14867 * 783421 * 9076901 * 556341246073

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Rebert wrote:

Q. Extension:

n  | a(n)     | prime factorization of concatenation a(1)|a(2)...a(n)

13 | 21179423 | 2135376432631715933766611741977155190124721179423 = 7^2 * 11 * 23 * 29 * 59 * 229 * 359 * 3299 * 6311 * 14867 * 783421 * 9076901 * 556341246073

14 | 31838299 | 213537643263171593376661174197715519012472117942331838299 = 3^3 * 19 * 47 * 53 * 127 * 233 * 257 * 331 * 1013 * 150379 * 337721 * 822067 * 2982345713 * 526311293257

 

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Emmanuel wrote:

The next term is  21179423.

The new concatenation is

2135376432631715933766611741977155190124721179423 =
 

(7^ 2}*11*23*29*59*229*359*3299*6311*14867*783421*9076901*556341246073.

 

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