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frugal numbers
A number  $n$  is called frugal if the number of digits in its prime factorization (including exponents greater than 1) is smaller than the number of digits of  $n$.

For example,  $2048=2^{11}$  is frugal.

The union of frugal and equidigital numbers gives economical numbers.

Some authors call the frugal numbers economical.

Every number greater than 1923 can be written as a sum of frugal numbers.

The smallest 3 × 3 magic square whose entries are frugal numbers is

128791725311421
123931385115309
162811044914823

The first frugal numbers are 125, 128, 243, 256, 343, 512, 625, 729, 1024, 1029, 1215, 1250, 1280, 1331, 1369, 1458, 1536, 1681, 1701, 1715 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
remainders of frugal numbers

A graph displaying how many frugal numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of frugal numbers
Useful links Wikipedia, Frugal number
R. G. E. Pinch, Economical numbers
The Prime Glossary, Economical Number
Frugal numbers can also be... (you may click on names or numbers and on + to get more values)

ABA 128 512 1024 + 999670898 aban 125 128 243 + 999000877 abundant 1280 1458 1536 + 50000000 Achilles 10125 10368 10976 + 999581472 admirable 15872 122624 128768 + 67100672 alternating 125 256 343 + 989870105 amenable 125 128 256 + 999999189 amicable 122368 18655744 134886465 169000448 apocalyptic 243 1280 1701 + 29929 arithmetic 125 343 1029 + 9999803 automorphic 625 90625 890625 + 212890625 binomial 117855 118341 130816 + 957053125 brilliant 1369 1681 1849 + 999002449 c.decagonal 119351 1987651 2122261 + 768862001 c.heptagonal 1331 1806848 2339072 + 678133681 c.nonagonal 130816 195625 1413721 + 957053125 c.octagonal 625 729 1369 + 999887641 c.pentagonal 22801 1254931 1363456 + 775060141 c.square 28561 325625 1041125 + 950698013 c.triangular 6241 27136 212629 + 651177091 cake 2048 40495625 75203584 + 488392192 Canada 125 compositorial 207360 696729600 congruent 125 343 1029 + 9998677 constructible 128 256 512 + 855638016 cube 125 343 512 + 997002999 Cunningham 33125 57122 101125 + 910953125 Curzon 125 729 1458 + 199998125 d-powerful 2048 5329 23763 + 9822259 de Polignac 14375 26875 27097 + 99977369 decagonal 1701 133407 469567 + 976515625 deceptive 237169 346029571 922350241 deficient 125 128 243 + 9999803 dig.balanced 625 3125 3721 + 199999501 double fact. 10321920 185794560 droll 192000 373248 640000 + 995328000 Duffinian 125 128 243 + 9999803 eban 32000 40000 50000 + 66000000 economical 125 128 243 + 20000000 emirpimes 1849 3481 3721 + 99460729 enlightened 256 2048 2560 + 761743661 eRAP 13213225 13595885 13638976 + 919657469 esthetic 343 76545 5456565 + 656543232 evil 125 243 343 + 999999189 fibodiv 240638283 721914849 Friedman 125 128 343 + 995328 gapful 1215 1250 1280 + 999937953 happy 1029 1215 1875 + 10000000 harmonic 33550336 Harshad 243 512 1215 + 999968571 heptagonal 16281 478953 589761 + 870740928 hex 32761 148519 1951327 + 680817481 hexagonal 117855 130816 195625 + 774821295 hoax 2401 10240 10375 + 99601831 Hogben 343 105301 179353 + 891589741 house 1175056 178825984 hyperperfect 176661 214273 1433701 + 925572421 iban 243 343 1024 + 777243 impolite 128 256 512 + 536870912 inconsummate 1536 8192 9375 + 991613 interprime 625 1215 1681 + 99865625 Jordan-Polya 128 256 512 + 995328000 junction 1024 1215 4913 + 99980041 Leyland 512 6250 93312 + 774840978 lucky 729 1029 1369 + 9985759 Lynch-Bell 128 13824 314928 1382976 magic 6826079 161414771 193710609 609093750 magnanimous 512 625 2209 + 22801 metadrome 125 128 256 + 134689 modest 14063 98441 112999 + 998016467 nialpdrome 10000 20000 32000 + 999887641 nonagonal 28125 218125 1365625 + 988621354 nude 128 1715 11664 + 499213368 oban 512 625 729 octagonal 22016 176661 196096 + 847392133 odious 128 256 512 + 999994891 palindromic 343 1331 10201 + 985383589 pancake 4096 120787 139129 + 931198591 pandigital 11045 12125 21870 + 379233148 pentagonal 585625 1572352 1628125 + 892222787 perfect 33550336 pernicious 625 1029 1250 + 9999803 plaindrome 125 128 256 + 555588999 Poulet 1194649 12327121 power 125 128 243 + 999887641 powerful 125 128 243 + 999887641 practical 128 256 512 + 10000000 prim.abundant 101632 102656 104704 + 81698816 pronic 261632 12841472 19127502 + 976531250 Proth 10625 14337 16129 + 999817217 pseudoperfect 1280 1458 1536 + 33550336 repunit 343 105301 179353 + 891589741 Rhonda 15625 1456611 1584117 + 922188518 Ruth-Aaron 125 1331 172773 + 978422247 Saint-Exupery 103680 245760 349920 + 983223840 self 512 1458 1715 + 999990625 self-describing 10311233 19322123 31331417 33311918 semiprime 1369 1681 1849 + 99460729 sliding 20000 25000 160625 + 925000000 Smith 729 10201 17496 + 99984843 square 256 625 729 + 999887641 star 11881 226981 1164241 + 381077581 strobogrammatic 6889 69169 1998661 + 196000961 super Niven 10000 20000 40000 + 999000000 super-d 1331 1369 1681 + 9992403 tau 128 625 1250 + 999981056 taxicab 134379 2991816 3628233 + 998951616 triangular 117855 118341 130816 + 957053125 tribonacci 10609 223317 trimorphic 125 625 4375 + 787109375 uban 1000000 2000000 2000033 + 98000000 Ulam 243 1536 1792 + 9998677 undulating 343 unprimeable 512 625 1715 + 10000000 untouchable 2048 10240 10368 + 979776 upside-down 8192 144669 151959 + 992357811 vampire 2187 108135 126846 + 75296875 Woodall 229375 Zuckerman 128 1715 11664 + 997411968 Zumkeller 1280 1536 1792 + 100000 zygodrome 1100000 1113344 1118866 + 999666111