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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Rahid Zaman wrote on July 11, 2026:

a(18) & a(19) = 436258209080137

436258209080137
436258209080143
436258209080153
436258209080191
436258209080207
436258209080237
436258209080273
436258209080299
436258209080327
436258209080341
436258209080357
436258209080429
436258209080431
436258209080467
436258209080489
436258209080501
436258209080509
436258209080513
436258209080531

Search exhaustively scanned from immediately after a(17), so there's none in between.

 

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