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panconsummate numbers
A number  $n$  is panconsummate if for every base  $b$, there is a number  $k$  such that  $k$  divided by its sum of digits in base  $b$  gives  $n$.

In other words, a number is panconsummate it is not inconsummate in any base.

Checking for this property is made easier by noting that a number  $n$  is always "consummate" in a base  $b\ge n$.

For example, 5 is panconsummate because (a)  $10=(1010)_2$  and  $10/(1+0+1+0)=5$, (b)  $10=(101)_3$  and  $10/(1+0+1)=5$, and (c)  $30=(132)_4$  and  $30/(1+3+2)=5$. On the contrary, 62 is not panconsummate because in base 10 it does not exist a number  $n$  such that  $n/\mathrm{sod}(n)$  = 62 (i.e., 62 is inconsummate).

The known panconsummate numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 18, 20, 21, 23, 24, 31, 34, 36, 37, 39, 40, 43, 45, 53, 54, 57, 59, 61, 69, 72, 73, 77, 78, 81, 85, 89, 91, 121, 127, 144, 166, 169, 211, 219, 231, 239, 257, 267, 271, 331, 337, 353, 361, 413, 481, 523, 571, 661, 721, 1093, 1291, 3097.

It there exist a further such number (which seems improbable), it must be greater than  $10^6$.

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

a-pointer 11 ABA 18 24 72 81 aban 10 11 12 + 721 abundant 12 18 20 + 144 Achilles 72 admirable 12 20 24 + 78 alternating 10 12 14 + 721 amenable 12 20 21 + 3097 apocalyptic 361 arithmetic 11 14 15 + 3097 astonishing 15 balanced p. 53 211 257 Bell 15 binomial 10 15 20 + 231 brilliant 10 14 15 + 481 c.decagonal 11 31 61 + 661 c.heptagonal 43 c.nonagonal 10 91 c.octagonal 81 121 169 361 c.pentagonal 31 331 c.square 61 85 481 c.triangular 10 31 85 + 571 cake 15 Catalan 14 Chen 11 23 31 + 1291 compositorial 24 congruent 14 15 20 + 1093 constructible 10 12 15 + 257 Cunningham 10 15 24 + 257 Curzon 14 18 21 + 413 cyclic 11 15 23 + 3097 D-number 15 21 39 + 267 d-powerful 24 43 89 267 de Polignac 127 331 337 decagonal 10 85 deceptive 91 481 deficient 10 11 14 + 3097 dig.balanced 10 11 12 + 721 double fact. 15 droll 72 Duffinian 21 36 39 + 3097 eban 34 36 40 54 economical 10 11 14 + 1291 emirp 31 37 73 337 emirpimes 15 39 85 + 3097 equidigital 10 11 14 + 1291 eRAP 20 24 esthetic 10 12 21 + 121 Eulerian 11 57 evil 10 12 15 + 1093 factorial 24 fibodiv 14 61 Fibonacci 21 34 89 144 Friedman 121 127 gapful 121 231 Gilda 78 good prime 11 37 53 + 331 happy 10 23 31 + 3097 Harshad 10 12 18 + 3097 heptagonal 18 34 81 hex 37 61 91 + 721 hexagonal 15 45 91 231 highly composite 12 24 36 hoax 85 166 361 Hogben 21 31 43 + 211 house 78 271 hungry 144 hyperperfect 21 iban 10 11 12 + 721 iccanobiF 39 idoneal 10 12 15 + 85 interprime 12 15 18 + 231 Jacobsthal 11 21 43 85 Jordan-Polya 12 24 36 + 144 junction 721 Kaprekar 45 katadrome 10 20 21 + 721 Kynea 23 Lehmer 15 85 91 + 3097 Leyland 54 57 lonely 23 53 211 Lucas 11 18 lucky 15 21 31 + 3097 Lynch-Bell 12 15 24 36 m-pointer 23 61 magic 15 34 magnanimous 11 12 14 + 661 metadrome 12 14 15 + 267 modest 23 39 59 + 721 Moran 18 21 45 + 3097 Motzkin 21 127 nialpdrome 10 11 20 + 721 nonagonal 24 nude 11 12 15 + 144 O'Halloran 12 20 36 oban 10 11 12 + 523 octagonal 21 40 481 odious 11 14 21 + 3097 palindromic 11 77 121 353 palprime 11 353 pancake 11 37 121 211 pandigital 11 15 21 78 partition 11 15 77 231 pentagonal 12 pernicious 10 11 12 + 3097 Perrin 10 12 39 Pierpont 37 73 257 plaindrome 11 12 14 + 337 power 36 81 121 + 361 powerful 36 72 81 + 361 practical 12 18 20 + 144 prim.abundant 12 18 20 78 prime 11 23 31 + 1291 primeval 37 pronic 12 20 72 Proth 57 81 257 + 481 pseudoperfect 12 18 20 + 144 repdigit 11 77 repfigit 14 61 repunit 15 21 31 + 1093 Ruth-Aaron 15 24 77 78 self 20 31 53 + 413 semiprime 10 14 15 + 3097 sliding 11 20 Smith 85 121 166 sphenic 78 231 square 36 81 121 + 361 star 37 73 121 + 1093 strobogrammatic 11 69 strong prime 11 37 59 + 331 super Niven 10 12 20 + 40 super-d 31 69 81 + 481 superabundant 12 24 36 tau 12 18 24 + 72 tetrahedral 10 20 tetranacci 15 triangular 10 15 21 + 231 tribonacci 24 81 trimorphic 24 truncatable prime 23 31 37 + 523 twin 11 31 43 + 1291 uban 10 11 12 + 91 Ulam 11 18 36 + 219 undulating 121 353 upside-down 37 73 91 wasteful 12 18 20 + 3097 weak prime 23 31 43 + 1291 Wieferich 1093 Woodall 23 Zuckerman 11 12 15 + 144 Zumkeller 12 20 24 + 78 zygodrome 11 77