Problems & Puzzles: Puzzles

Puzzle 938. The Soothsayer sequence

G. L. Honaker, Jr. sent the following nice puzzle:

The soothsayer sequence: An n-digit number whose digit(s) describe the number of distinct prime factors in each of the following n integers. The sequence begins with 0, 1, 12, 21, 22422, 24223, 33333, 34441524, 4242436235, 23443535352, 34462443242, 35256523324, 4341535435353, 4645441523344, 5244526446515, 5335524234335, ...

The last 7 terms are from Giovanni Resta.

Explanation:

22422 is in the sequence because:

 22423 17*1319 2 Distinct prime factors 22424 2^3*2803 2 Distinct prime factors 22425 3*5^2*13*23 4 Distinct prime factors 22426 2*11213 2 Distinct prime factors 22427 41*547 2 Distinct prime factors

Carlos Rivera switched a little bit the original definition now counting all the primes with multiplicity.

The soothsayer sequence-II: An n-digit number whose digit(s) describe the number of prime factors with multiplicity in each of the following n integers. The sequence begins 0, 1, 12, 21, 4224, 153426, 442451, 471614, 523291, 4336232, 474335342, 3624263478, ... .

4224 is in the sequence because:

 4225 5^2*13^2 4 Distinct prime factors 4226 2*2113 2 Distinct prime factors 4227 3*1409 2 Distinct prime factors 4228 2^2*7*151 4 Distinct prime factors

Q. Extend both sequences.

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