Problems & Puzzles: Puzzles

 Puzzle 822. About OEIS sequence A193890 Arkadiusz Wesolowski  sent the following nice puzzle: In sequence http://oeis.org/A193890 I am interested in finding primes p such that replacing any single decimal digit d with 3*d produces another prime. The sequence starts as follows: 11, 311, 1301, 10133, 1030031.   Q1. What is the next term? Q2. Can you find any other primes that belong to this sequence? Q3. Are there any such 19-digit prime numbers? (Happy hunting!)

Contributions came from Jan van Delden, Dmitry Kamenetsky.

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Jan wrote:

The next term is larger than 10^22.

These numbers have the property that:
I)      The digits 1 and 2 are not present simultaneously.
II)     The number of digits equal to 1 must be 2 mod 3.
III)    The number of digits equal to 2 must be 2 mod 3.

I checked for prime numbers where (in the same fashion) a digit d is replaced by 7*d:

Solutions 11, 101. The next solution is larger than 10^31.

I checked for prime numbers where (in the same fashion) a digit d is replaced by 9*d:

The first solution is larger than 10^31. Is there any?

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Dmitry wrote:

I cannot find any new primes that belong to A193890. I have searched:

1. all numbers up to 740 digits with 2 or 3 non-zero digits
and
2. all numbers up to 330 digits with 4 non-zero digits

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