Problems & Puzzles: Puzzles

Puzzle 414. Prime lucky number triplets

Steven Harvey proposes the following interesting puzzle:

For sure you know what a lucky number is.

Reviewing a list of them it appears that the first three consecutive lucky numbers which are also prime numbers are 2221, 2239, and 2251.  The next three consecutive lucky numbers are 3301, 3307 and 3313, which are not only consecutive lucky numbers but consecutive prime numbers also.  The next three consecutive prime lucky numbers are 5839, 5851, and 5869, and I there are no more in the first thousand.

Note. I [CR] found 3 quartets:

9631, 43, 49, 61 (consecutive primes too!)
31033, 39, 51, 69
71329, 41, 47, 59

So the puzzle is:

Q1. Are there k-tuplets for k>4?

Q2.  Slightly related, is there a luckiness test, similar to a primality test?

Q3. I [CR] found a large series of 48 consecutive composite & lucky numbers (from 73419 to 74059). Can you find a larger series than this?

 


 

 

Fred Schneider wrote:

I only checked through 750K but I found:

Longer composite Streaks
287691 to 288501 : 58
562039 to 563065 : 58
583983 to 585027 : 67

First case of 5 consecutive lucky numbers that are prime (These primes
aren't consecutive)
162451, 162457, 162493, 162523, 162529
 

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