Problem 85. Follow-up to puzzles 18 and 31

On July 5, 2023, Paul W. Dyson sent solutions to Q2 of my old Puzzle 18, (and Puzzle 31).

He wrote:

"Last year I ran a program for about a month and a half on a RTX3060Ti GPU to find the next solution for S(pk) % pk = 0. This is Puzzle 18 question 2 on your Prime Puzzles website. The next value is pk = 55,691,042,365,834,801. This sequence is OEIS A007506.

The code was also searching for the next solution of S(pk) % k = 0 (i.e. % k, rather than % pk). So I kept it running until I found a solution to that one too. After a total of about 5 and a half months it found k = 6,361,476,515,268,337. This sequence is OEIS A045345.

I kept looking for S(pk) % pk = 0 for the whole time, and found that the next solution must be greater than pk = 253,814,097,223,614,463"

In short he found a(6) for A007506 and a(16) for A045345. After a big clap for his amazing results...

Q. Find a(7) for A007506 and a(17) for A(045345)