"Base $b$ valuated prime" is an odd prime, whose representations in all natural number bases $2,3,\dots,b$ are also prime when evaluated as base $b$ numbers.
That is, write the (prime) number in bases $2,3,\dots,b$. Then read those expressions as base $b$ expressions, to obtain new numbers, which all must be prime to satisfy the definition.
Here is a table of first ten examples, for base $b=2,3,\dots,10$ valuated primes:
2: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...
3: 7, 11, 13, 41, 47, 67, 73, 79, 109, 127, ...
4: 43, 61, 139, 151, 229, 277, 379, 439, 787, 823, ...
5: 41, 151, 181, 271, 1759, 2011, 3061, 5869, 9601, 14591, ...
6: 45481, 102181, 202621, 229261, 275881, 498301, 570421, 894541, 906541, 1103101, ...
7: 701, 805583, 3550733, 29725363, 39296861, 44098673, 50976553, 55402397, 60343249, 62351203, ...
8: 65484343, ?, ?, ?, ?, ?, ?, ?, ?, ?, ...
9: ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ...
10: ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ...
For $b=2$, we have the sequences of all primes themselves.
For $b=3$, they are in the OEIS sequence $A235266$, which links to similar sequences.
- For $b\ge 4$, I haven't found similar sequences or relevant information.
The ? stands in places where I didn't find all ten examples.
For $b=8,9,10$, I checked first $10^8$ primes. - But, the examples are probably much larger?
Can these primes exits in all $b$? (Can we find second $b=8$ or first $b=9,10$ examples?)
Can we estimate the number of such $b$ valuated primes below some upper bound?
Roughly speaking, we are taking some prime $p_n$, and evaluating numbers of $\approx$ sizes:
$$b\land\{\log_2p_n\},b\land\{\log_3p_n\},\dots,b\land\{\log_bp_n\}$$
Which need to be all simultaneously prime. (Where $\land$ is exponent, for readability).
Can we test for these numbers faster (Can we establish any useful characterizations)?
This question was motivated by a recent similar question.