Economical Number

A number  is called an economical number if the number of digits in the prime factorization of  (including powers) uses fewer digits than the number of digits in . The first few economical numbers are 125, 128, 243, 256, 343, 512, 625, 729, ... (OEIS A046759). Pinch shows that, under a plausible hypothesis related to the twin prime conjecture, there are arbitrarily long sequences of consecutive economical numbers, and exhibits such a sequence of length nine starting at 1034429177995381247.

REFERENCES:

Hess, R. I. "Solution to Problem 2204(b)." J. Recr. Math. 28, 67, 1996-1997.

Pinch, R. G. E. "Economical Numbers." https://www.chalcedon.demon.co.uk/publish.html#62.

Rivera, C. "Problems & Puzzles: Puzzle 053-Sequences of Consecutive Economical Numbers." https://www.primepuzzles.net/puzzles/puzz_053.htm.

Santos, B. R. "Problem 2204. Equidigital Representation." J. Recr. Math. 27, 58-59, 1995.

Sloane, N. J. A. Sequence A046759 in "The On-Line Encyclopedia of Integer Sequences."