d-powerful numbers

An integer $n$

is called digitally powerful (here d-powerful) if it can be expressed as a sum of positive powers of its digits.

For example,

$3459872 = 3^1 + 4^6 + 5^5 + 9^6 + 8^3 + 7^7 + 2^{21}$

is d-powerful.

Carlos Rivera (see link), who allows zero exponents, calls there numbers handsome.

The first d-powerful numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 43, 63, 89, 132, 135, 153, 175, 209, 224, 226, 262, 264, 267, 283, 332, 333, 334, 357, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 407, 445, 463, 518, 598, 629, 739, 794, 849, 935, 994 more terms

You can download a text file (d-powerful_up_1e6.txt, 0.9 MB), with a list of the 30067 d-powerful numbers up to $10^6$ and the corresponding sums of powers.

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

A graph displaying how many d-powerful numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links OEIS, Sequence A007532
Carlos Rivera, Puzzle 15. Narcissistic and Handsome Primes

D-powerful numbers can also be... (you may click on names or numbers and on + to get more values)

ABA 24 375 2048 + 9954722 aban 24 43 63 + 7000895 abundant 24 132 224 + 9999752 Achilles 1323 5324 8712 + 9279432 admirable 24 224 1074 + 9996234 alternating 43 63 89 + 898729 amenable 24 89 132 + 9999865 amicable 783556 879712 2728726 + 7684672 apocalyptic 224 226 994 + 29967 arithmetic 43 89 132 + 9999865 astonishing 2132532 automorphic 376 balanced p. 373 17483 22447 + 9923477 binomial 153 378 1326 + 9792525 brilliant 209 377 407 + 9988337 c.decagonal 8201 249761 258781 + 4394531 c.heptagonal 43 463 2843 + 9442322 c.nonagonal 16471 24976 32896 + 7882435 c.octagonal 4225 5329 49729 + 8323225 c.pentagonal 226 3706 4951 + 9394456 c.square 2245 9385 24865 + 9843485 c.triangular 1306 2224 3385 + 9734635 cake 378 2048 2626 + 7207552 Carmichael 2465 278545 3146221 Catalan 132 Chen 89 379 2179 + 9979337 compositorial 24 congruent 24 63 135 + 9999423 constructible 24 1542 2048 + 7864320 cube 5832 35937 287496 + 4657463 Cullen 2228225 Cunningham 24 63 224 + 9865882 Curzon 89 153 209 + 9994326 cyclic 43 89 209 + 9999827 D-number 63 267 849 + 7040589 de Polignac 373 2203 2263 + 9994657 decagonal 175 370 2626 + 9457162 deceptive 231337 2358533 2392993 + 9567673 deficient 43 63 89 + 9999865 dig.balanced 135 153 209 + 9999827 droll 2240 4224 39424 Duffinian 63 175 209 + 9999865 eban 2062 2064 32052 + 4052036 economical 43 89 135 + 9997263 emirp 739 3257 3527 + 9984827 emirpimes 226 629 794 + 9992722 enlightened 2048 2238728 equidigital 43 89 135 + 9997263 eRAP 24 56563 78524 + 6275378 esthetic 43 89 1234 + 876543 evil 24 43 63 + 9999827 factorial 24 fibodiv 2733 24712 29923 + 209987 Fibonacci 89 377 Friedman 153 1255 2048 + 995346 frugal 2048 5329 23763 + 9822259 gapful 132 135 264 + 9999827 Gilda 65676 4848955 good prime 739 2063 2203 + 9887573 happy 226 262 376 + 9998724 Harshad 24 63 132 + 9998252 heptagonal 2205 2356 3940 + 9591264 hex 4447 22447 23497 + 9682237 hexagonal 153 378 1326 + 9792525 highly composite 24 hoax 1255 2064 2227 + 9992722 Hogben 43 463 2353 + 9526483 Honaker 2803 4153 4397 + 9972233 house 874234 2596375 9272286 hyperperfect 2133 10693 214273 296341 iban 24 43 224 + 777322 idoneal 24 357 impolite 2048 inconsummate 63 371 372 + 997784 interprime 334 370 376 + 9999227 Jacobsthal 43 2796203 Jordan-Polya 24 2048 24576 + 7864320 junction 1306 2315 2517 + 9997442 Kaprekar 2223 4950 356643 9372385 katadrome 43 63 5320 + 9876532 Lehmer 2465 3145 8245 + 9828295 Leyland 4240 262468 2097593 lonely 2179 1357265 4652430 Lucas 1364 lucky 43 63 135 + 9996789 Lynch-Bell 24 132 135 + 2189376 m-pointer 2131 magic 175 2465 66351 + 5848655 magnanimous 43 89 209 + 4488245 metadrome 24 89 135 + 2456789 modest 89 209 333 + 9865423 Moran 63 153 209 + 9998252 Motzkin 2356779 narcissistic 153 370 371 + 9926315 nialpdrome 43 63 332 + 9999865 nonagonal 24 4959 7944 + 8978409 nude 24 132 135 + 9983232 oban 63 89 333 + 935 octagonal 2133 2465 4485 + 8923425 odious 224 262 283 + 9999865 Ormiston 135497 235997 327779 + 9965479 palindromic 262 333 373 + 9859589 palprime 373 98389 3223223 + 9749479 pancake 379 407 4951 + 9642637 panconsummate 24 43 89 267 pandigital 135 1634 2134 + 9977513 partition 135 1255 2679689 pentagonal 376 3432 3725 + 9975572 pernicious 24 132 224 + 9999865 Perrin 209 2627 24914 Pierpont 3457 plaindrome 24 89 135 + 6667789 Poulet 2465 194221 223345 + 9567673 power 2048 4225 4624 + 9972964 powerful 1323 2048 4225 + 9972964 practical 24 132 224 + 9997668 prim.abundant 1074 1542 2205 + 9996234 prime 43 89 283 + 9993623 pronic 132 2352 6972 + 9837632 Proth 209 2241 2753 + 9875457 pseudoperfect 24 132 224 + 999252 repdigit 333 repfigit 62662 repunit 43 63 463 + 9573285 Rhonda 5832 196355 226953 + 9892552 Ruth-Aaron 24 153 370 + 9486722 Saint-Exupery 267540 self 132 209 334 + 9999752 semiprime 209 226 262 + 9999827 sliding 2225 22025 62660 + 3231250 Smith 378 1255 2227 + 9992762 Sophie Germain 89 2063 2753 + 9978833 sphenic 357 370 374 + 9999865 square 4225 4624 5329 + 9972964 star 2773 23437 26533 + 9563437 straight-line 135 333 357 + 876543 strong prime 379 739 2063 + 9993623 super Niven 24 8200 42240 262200 super-d 332 333 334 + 9995572 superabundant 24 tau 24 132 372 + 9995424 taxicab 32832 46683 525824 + 4624776 tetrahedral 366145 562475 657359 + 6435689 tetranacci 39648 2033628 triangular 153 378 1326 + 9792525 tribonacci 24 trimorphic 24 375 376 + 59375 truncatable prime 43 283 373 + 9979337 twin 43 283 463 + 9972583 uban 43 63 89 Ulam 175 209 370 + 9998724 undulating 262 373 2626 + 43434 unprimeable 518 1074 1672 + 9999752 untouchable 262 372 518 + 996732 upside-down 357 2738 4736 + 9785231 vampire 193257 263074 284598 729688 wasteful 24 63 132 + 9999865 weak prime 43 89 283 + 9979643 weakly prime 3326489 weird 9272 295330 338572 + 876964 Wieferich 22953 Woodall 63 4373 2359295 Zeisel 4293793 5069629 Zuckerman 24 132 135 + 6422976 Zumkeller 24 132 224 + 99402 zygodrome 333 2244 22333 + 9933222