Problems & Puzzles: Puzzles

 Puzzle 385. Follow up to Puzzle 104 Giovanni Resta recently wrote, as a follow up to Puzzle 104: 333366670001 = 3333^3 + 6667^3 + 0001^3 Another infinite family is that which contains numbers like [3][7][1], [33][67][01], [333][667][001], where a(k) will have 3*k digits. I'v checked a(n) for n<1000 and it is prime for n=3, 4 (as we already know) and for n=46, which will provide a nice, certified, prime of 46*3 = 138 digits. Questions. 1. Can you find the next (n>46) candidate to prime of this kind? 2. Is there another infinite family like the found by Resta containing potential primes?

Contribution came from Farideh Firoozbakht:

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Farideh wrote:

1. The next term is greater than 8620, so the prime corresponding to
next term has more than 25860 digits.

2. I didn't find such family of primes.

We can extend the sequence A056733:

153, 370, 371, 407, 165033, 221859, 336700,
336701, 340067, 341067, 407000, 407001, 444664, 487215,
982827, 983221, 166500333, 296584415, 333667000, 333667001,
334000667, 710656413, 828538472, ...

to: Each number is sum of the cubes of its 3 sections where
the sections for each number can have distinct lengths.

153, 370, 371, 407, 1000, 1001, 2213, 4160, 4161,
41833, 165033, 221859, 336700, 336701, 340067,
340067, 341067, 407000, 407001, 444664, 487215,
684045, 982827, 983221, 1000000, 1000000, 1000000,
1000001, 1000001, 1000001, 1000407, 1001000, 1001000,
1001001, 1001001, 1950103,...

2213 = 2^3+2^3+13^3
1000 = 10^3+0^3+0^3
41833 = 4^3+18^3+33^3
1950103 = 1^3+95^3+0103^3
...

I didn't find the request pattern for this sequence.

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