Problems & Puzzles: Puzzles

Puzzle 1002. Consecutive primes that are also happy numbers Carlos Rivera computed the smallest start of a run of k consecutive primes that are also happy integers.   Here are his results (See A334982)   7, 19, 3067, 3313, 3313, 11550481, 11550481, 475389133, 1128863117, 1175267983   Example: For k=2, thee smallest start of o run of 2 consecutive primes and happy integers is 19, because 19 & 23 are consecutive primes and happy integers.   Q. Can you extend this sequence?

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During the week 24-30 of May, 2020, contributions came from Giovanni Resta, Emmanuel Vantieghem, Jan van Delden, Oscar Volpatti.

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Giovanni wrote:

These are the further terms I found:

11 19507818293
12 176622893141
13 176622893141
14 495495126109
15 4411512992681
16 122634868247173
17 122634868247173

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Emmanuel wrote:

The next term in your sequence is  19507818293.

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Jan wrote:

Happy anniversary. My contribution:

 

A list which gives the smallest prime that starts a sequence of maximal length k.

This will give slightly more information than what was asked for.

 

Length: 1 Start prime 7

Length: 2 Start prime 19

Length: 3 Start prime 3067

Length: 5 Start prime 3313

Length: 4 Start prime 6329

Length: 7 Start prime 11550481

Length: 6 Start prime 15676597

Length: 8 Start prime 475389133

Length: 9 Start prime 1128863117

Length: 10 Start prime 1175267983

Length: 11 Start prime 19507818293

Length: 13 Start prime 176622893141

Length: 12 Start prime 229543158851

Length: 14 Start prime 495495126109

Length: 15 Start prime 4411512992681

 

And no longer sequence for p<4.1*10^13

 

For odd numbers less than 10^10 the distribution of

m: length of consecutive happy numbers, is as follows:

 

m    Frequency

0 4251740067

1   577014799

2     69506265

3       8735432

4       1503889

5                    0

6             1792

 

The fraction of odd happy numbers is 0.149652.

 

First occurrences of sequences with a given length:

 

Length: 1 Start 1

Length: 2 Start 291

Length: 4 Start 1333

Length: 3 Start 2899

Length: 6 Start 3313

 

It is harder to find sequences of consecutive odd numbers that are happy than consecutive happy prime numbers.

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Oscar wrote:

I found four more terms:

a(11) = 19507818293;

a(12) = 176622893141;

a(13) = 176622893141;

a(14) = 495495126109.

Next sequence of 14 consecutive happy primes starts at p = 1693652294143.

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