25777...77777 (398-digits)
This number is a probable-prime (likely to be a prime but we have not (re-)proven it on this site).
25777777 7777777777 7777777777 7777777777 7777777777
7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777
7777777777 7777777777 7777777775 7777777777 7777777777 7777777777 7777777777
7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777
7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777
7777777777 7777777777 7777777777 7777777777 7777777777 7777777777 7777777777
This prime-digit prime has sum of digits equal to 2777, and 2777 has sum of digits equal to 23, and 23 has sum of digits equal to 5. It is the smallest prime-digit prime with this property, if all primes in the sequence are restricted to prime-digit primes, where the final iterated sum of digits is 5. The prime-digit prime for the next term of this sequence will be outrageously gigantic. [Sumpeloh]
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