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

For example,  $3072=3\cdot2^{10}$  is equidigital.

The union of equidigital and frugal numbers gives economical numbers.

The first equidigital numbers are 2, 3, 5, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 35, 37, 41, 43, 47, 49, 53, 59, 61, 64 more terms

The earliest runs of exactly  $n=1,2,\dots,12$  equidigital numbers start at 5, 10, 1, 191, 13, 1134, 157, 1169312, 100149391, 1016258233, 107375450119, and 10271593487494, respectively.

The smallest 3 × 3 magic square whose entries are consecutive equidigital numers is

125691255512565
125591256312567
125611257112557

Below, the spiral pattern of equidigital numbers up to  $200^2$. See the page on prime numbers for an explanation and links to similar pictures.

spiral pattern of equidigital numbers

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

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

a-pointer 11 13 101 + 19993951 ABA 32 64 81 + 19983842 aban 10 11 13 + 19000999 abundant 112 160 162 + 19999488 Achilles 1125 1152 1323 + 19961856 admirable 224 1372 1504 + 8382464 alt.fact. 19 101 619 + 3301819 alternating 10 14 16 + 18989878 amenable 13 16 17 + 19999985 amicable 1184 2090656 9437056 + 19154336 apocalyptic 157 192 224 + 30000 arithmetic 11 13 14 + 9999991 astonishing 15 27 1863 automorphic 25 109376 7109376 balanced p. 53 157 173 + 19999981 Bell 15 203 877 115975 bemirp 1061 1091 1601 + 19986091 binomial 10 15 21 + 19961721 brilliant 10 14 15 + 19999979 c.decagonal 11 31 61 + 19990001 c.heptagonal 43 71 106 + 19967263 c.nonagonal 10 1081 1225 + 19911205 c.octagonal 25 49 81 + 19936225 c.pentagonal 16 31 106 + 19930381 c.square 13 25 41 + 19964881 c.triangular 10 19 31 + 19989226 cake 15 64 1351 + 19250624 Canada 581 16999 Carmichael 10585 15841 1193221 + 19384289 Carol 47 223 3967 + 16769023 Catalan 14 Chen 11 13 17 + 19999739 compositorial 192 1728 17280 2903040 congruent 13 14 15 + 9999991 constructible 10 15 16 + 17895424 cube 27 64 1000 + 8489664 Cullen 25 161 2049 1048577 Cunningham 10 15 17 + 19980901 Curzon 14 21 29 + 19999953 cyclic 11 13 15 + 9999991 D-number 15 21 111 + 2999949 d-powerful 43 89 135 + 9997263 de Polignac 127 149 251 + 19999963 decagonal 10 27 175 + 19582837 deceptive 259 4187 10001 + 19965037 deficient 10 11 13 + 9999991 dig.balanced 10 11 15 + 16771072 double fact. 15 105 384 3840 droll 2240 5184 6272 + 17842176 Duffinian 16 21 25 + 9999625 eban 32 64 2000 + 6064000 economical 10 11 13 + 19999999 emirp 13 17 31 + 19999981 emirpimes 15 49 115 + 19999979 enlightened 250 2176 2304 + 5117695 eRAP 7625 68479 162810 + 19312223 esthetic 10 21 23 + 12343454 Eulerian 11 1013 1191 + 13824739 evil 10 15 17 + 19999985 factorial 362880 3628800 fibodiv 14 19 47 + 19999999 Fibonacci 13 21 89 + 1346269 Friedman 25 121 127 + 996543 gapful 105 121 135 + 19999750 Gilda 29 49 683 + 4848955 good prime 11 17 29 + 19988359 happy 10 13 19 + 9999991 Harshad 10 21 27 + 19999709 heptagonal 81 112 189 + 19792269 hex 19 37 61 + 19945987 hexagonal 15 1225 1431 + 19955403 hoax 160 166 250 + 19999858 Hogben 13 21 31 + 19994313 Honaker 131 263 457 + 19984511 house 32 155 271 + 16838677 hungry 17 2003 161449 2401519 hyperperfect 21 301 1333 + 18116737 iban 10 11 14 + 777743 iccanobiF 13 1053 4139 idoneal 10 13 15 + 177 impolite 16 32 64 inconsummate 161 173 371 + 999953 insolite 111 1122112 interprime 15 21 64 + 19999858 Jacobsthal 11 21 43 + 2796203 Jordan-Polya 16 32 64 + 19353600 junction 101 103 105 + 19999946 Kaprekar 17344 148149 katadrome 10 21 31 + 9875321 Kynea 23 79 287 + 16785407 Lehmer 15 133 259 + 19981237 Leyland 17 32 145 + 16777792 lonely 23 53 211 + 12623189 Lucas 11 29 47 + 12752043 lucky 13 15 21 + 9999823 Lynch-Bell 15 135 162 + 1687392 m-pointer 23 61 1123 + 19122211 magic 15 111 175 + 18797855 magnanimous 11 14 16 + 8608081 metadrome 13 14 15 + 13456789 modest 13 19 23 + 19999999 Moran 21 27 111 + 19999709 Motzkin 21 127 15511 narcissistic 371 nialpdrome 10 11 21 + 9999991 nonagonal 111 1216 1491 + 19769509 nude 11 15 111 + 19999791 oban 10 11 13 + 997 octagonal 21 133 1281 + 19979521 odious 11 13 14 + 19999999 Ormiston 1913 1931 18379 + 19996297 palindromic 11 101 111 + 18800881 palprime 11 101 131 + 9989899 pancake 11 16 29 + 19999651 panconsummate 10 11 14 + 1291 pandigital 11 15 19 + 16434817 partition 11 15 101 + 10619863 pentagonal 35 145 287 + 19927215 pernicious 10 11 13 + 9999991 Perrin 10 17 29 + 16042867 Pierpont 13 17 19 + 19131877 plaindrome 11 13 14 + 19999999 Poulet 1387 2047 2701 + 19985269 power 16 25 27 + 19998784 powerful 16 25 27 + 19998784 practical 16 32 64 + 9999936 prim.abundant 1184 1312 1376 + 19991936 prime 11 13 17 + 19999999 primeval 13 37 107 + 12345679 pronic 16256 16512 65792 + 19514306 Proth 13 17 25 + 19996673 pseudoperfect 112 160 162 + 998816 repdigit 11 111 11111 1111111 repfigit 14 19 47 + 1084051 repunit 13 15 21 + 19994313 Rhonda 1568 2835 5832 + 18552839 Ruth-Aaron 15 16 25 + 19968209 Saint-Exupery 3840 12960 30720 + 19710540 Sastry 183 self 31 53 64 + 19999999 self-describing 10123133 10153133 10153331 + 19333115 semiprime 10 14 15 + 19999982 sliding 11 25 29 + 16265000 Smith 27 121 166 + 19999858 Sophie Germain 11 23 29 + 19999379 sphenic 105 1005 1015 + 19999985 square 16 25 49 + 19998784 star 13 37 73 + 19994701 straight-line 111 123 135 + 1234567 strobogrammatic 11 101 111 + 19996661 strong prime 11 17 29 + 19999999 subfactorial 265 1334961 super Niven 10 1000 2000 + 15300000 super-d 19 31 81 + 9999931 tau 384 448 640 + 19999360 taxicab 110808 149389 195841 + 18673625 tetrahedral 10 35 12341 + 19131795 tetranacci 15 29 401 + 14564533 triangular 10 15 21 + 19961721 tribonacci 13 81 149 + 15902591 trimorphic 25 49 249 + 12890624 truncatable prime 13 17 23 + 19981373 twin 11 13 17 + 19999549 uban 10 11 13 + 19000099 Ulam 11 13 16 + 10000241 undulating 101 121 131 + 5656565 unprimeable 320 896 1072 + 9999625 untouchable 146 162 448 + 998144 upside-down 19 37 64 + 19982119 vampire 1435 110758 117067 + 19554277 weak prime 13 19 23 + 19999927 weakly prime 294001 505447 584141 + 19851991 Wieferich 1093 3511 10533 17555 Woodall 17 23 159 + 14680063 Zeisel 105 233569 1073305 + 18175361 Zuckerman 11 15 111 + 19919196 Zumkeller 112 160 192 + 99584 zygodrome 11 111 6655 + 11999933