In both sequences p(i) represents the i-th prime.

Solution:

Contribution came from Sambit Nayak and Jim Fougeron:

I tested all primes (> 2475989), up to the 200,000,001 th prime. (The 200,000,001 th prime is 4,222,234,763.) But I could not find any prime satisfying the property of the sequence. Based on these results, I guess that there is no other term in the sequence.

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Faride reports the following:

Next term of the sequence A097223 (17,73,2475989,...) is greater than 12207648773 and the next term of the sequence A097227 (14,154,1196,279174,...) is greater than 11400000000.

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Jim Fougeron found no solution for part 1) up to 12B and for part 2) up to 10^64. Here is his email:

I checked part a up to about 12B. No good values found.

For part b, I started a "slow" search, then realized that I could speed things up significantly. My method is this. First, we are simply looking for "good" permutations of mixed base numbers. I will show (in decimal) the pattern that gets all usable permutations:

1111
1112
1113
...
1119
1122
1123
1124
...
1129
1133
1134
...
1199
1222
1223
...
9999

As you can see, 0's are not used (they are invalid). Also, we skip from 1119 to 1122. This is because 1120 has a zero. 1121 is 1112, and thus a dup. Also, keep in mind that the decimal representations shown above (say 1123) are actually in base prime, this 1123 is 2*2*3*5 which we will be computing, and converting back to base 10.

Using the above method, I created a program which would start from the 1111.....111 value, and compute this in base-prime (vs in base 10). I then created a function that is sent a "base" digit, and the result digit, and the would update the "base-prime" number. Thus, if computing the current number, 33355556 (which is 5^3*11^4*13), the next number would be 3335557, I would simply call the function passing it 13 and 17. Thus 13 is divided out, then 17 is multiplied in (since p(6)=13 and p(7)=17).

At each step, this "base-prime" number is converted into a base-10 number. This was only done if the number was "in range" (i.e. had the same number of decimal digits as we are using. Thus, if the base-10 number of 5^3*11^4*17 contained the digits 33355556, (it is 31112125) then we have success. These digits can be in any order, but ALL must be accounted for, with no extras. A quick check is during the base-10 conversion, if there are any '0' digits, then abort, as there can be no 0's due to p(0) not making since.

I have checked up to 9999999..999*10^64 with no successful values. The 10^64 stopping point was due to the time the program was taking to run, and due to I was using a 64 bit integer to track which digits were used, while searching between the original "permutation" base-10, and the base-10 from the base-prime to base-10 conversion. I could easily write the program to handle more than 64 digits, but I  imagine this puzzle will be very hard to find the next value. That value might have 1000's of digits (just a wild guess, as it could have only 65 digits :)

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