What's so special about 8466772177^34?

When I introduced my factorization balancing act last year, I gave the example of a large (13-balanced) factorization integer:

7^85*918679^8   <120>   k=13

The number's 120-digit decimal expansion joined with the 10 digits of its normally-expressed factorization contains altogether exactly 13 each of the digits zero through nine. I suggested that it hadn't been too difficult to obtain — essentially, brute-force searching the product of a prime to a power and another prime to a power. In spite of this number's relatively easy generation, I hadn't found a larger example until recently:

8466772177^34   <338>   k=35

That is, the 338-digit integer joined with the 12 digits of its factorization contains exactly 35 each of the ten decimal digits!