Problems & Puzzles: Puzzles

Puzzle 53.- Sequences of consecutive economical numbers.

A number E is called (after B. Recam) an economical number, if the expression of E as a product of powered primes uses fewer digits than the digits of E.

The first 8 economical numbers are:

125, 128, 243, 256, 343, 512, 625, 729, …. (See Neils sequence A046759)

125 =5^3, 128 =2^7, etceteras.

Do exist consecutive economical numbers? The answer is yes. Here we are interested only in the earliest sequences of K=>2 consecutive economical numbers. I have found the earliest sequences for K = 2, 3 and 4:

K=2
4374 = 2* 3^ 7
4375 = 5^ 4* 7

K=3
1097873 = 7* 47^ 2* 71
1097874 = 2* 3^ 7* 251
1097875 = 5^ 3* 878

K=4 (see also A047738)
179210312 = 2^ 3* 4733^ 2
179210313 = 3^ 5* 97* 7603
179210314 = 2* 29* 37^ 3* 61
179210315 = 5* 79^ 2* 5743

Find the earliest sequence of 5<=K<=10 consecutive economical numbers.