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inconsummate numbers
Conway called  $n$  a inconsummate number if there is not a number  $k$  which divided by its sum of digits gives  $n$.

For example, 47 is not inconsummate because  $846/(8 + 4 + 6) = 47$. On the contrary, 62 is inconsummate because it does not exist a number  $n$ such that  $n/\mathrm{sod}(n)$  = 62.

A related family is that of panconsummate numbers, i.e., those numbers which are not inconsummate in any base.

The following table reports the number of inconsummate numbers up to  $10^k$  for  $k=1,2,\dots,9$.

up to101102103 104105106 107108109
#0 6111143716430183089190528518907944183706706

The first inconsummate numbers are 62, 63, 65, 75, 84, 95, 161, 173, 195, 216, 261, 266, 272, 276, 326, 371, 372 more terms

Below, the spiral pattern of inconsummate numbers up to  $250^2$. See the page on prime numbers for an explanation and links to similar pictures.

spiral pattern of inconsummate numbers

The smallest 3 × 3 magic square whose entries are consecutive inconsummate numbers is

289956289947289953
289949289952289955
289951289957289948

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

a-pointer 3041 3221 5153 + 995801 ABA 882 1536 2592 + 999698 aban 62 63 65 + 993 abundant 84 216 272 + 999980 Achilles 432 675 972 + 996872 admirable 84 426 438 + 999858 alt.fact. 326981 alternating 63 65 161 + 989892 amenable 65 84 161 + 999993 amicable 5564 10856 66992 + 980984 apocalyptic 411 443 497 + 29995 arithmetic 62 65 95 + 999995 astonishing 216 3591 20328 21762 balanced p. 173 563 3307 + 993869 Bell 21147 bemirp 1601 16001 betrothed 75 195 16587 + 549219 binomial 84 276 630 + 980700 brilliant 377 493 527 + 968087 c.decagonal 3251 7031 8201 + 979031 c.heptagonal 386 2276 3046 + 970208 c.octagonal 441 1521 2601 + 962361 c.pentagonal 276 1266 1626 + 968766 c.square 761 3281 4705 + 989825 c.triangular 30460 108139 237010 + 736051 cake 834 4526 5489 + 893376 Canada 8549 Catalan 4862 Chen 431 443 461 + 999149 congruent 62 63 65 + 999978 constructible 272 771 816 + 983055 cube 216 9261 35937 + 804357 Cullen 65 161 2049 Cunningham 63 65 195 + 994008 Curzon 65 173 261 + 999978 cyclic 65 95 161 + 999995 D-number 63 195 381 + 999993 d-powerful 63 371 372 + 997784 de Polignac 1271 1541 1649 + 999931 decagonal 5076 10251 18837 + 990522 deceptive 4187 145181 166499 + 981317 deficient 62 63 65 + 999995 dig.balanced 75 195 216 + 999993 double fact. 945 droll 9984 451584 537600 718848 Duffinian 63 65 75 + 999993 eban 62 2042 2046 + 66062 economical 161 173 371 + 999953 emirp 761 941 971 + 999931 emirpimes 62 326 381 + 999911 enlightened 236196 290912 equidigital 161 173 371 + 999953 eRAP 4233 25592 38180 + 991248 esthetic 65 432 543 + 898787 Eulerian 4293 262125 evil 63 65 75 + 999993 factorial 362880 fibodiv 75 488 497 + 974994 Fibonacci 377 317811 514229 Friedman 216 2592 3281 + 996543 frugal 1536 8192 9375 + 991613 gapful 195 1037 1053 + 999936 Gilda 2207 5346 30254 + 933138 Giuga 66198 good prime 431 563 821 + 999953 happy 326 383 386 + 999938 harmonic 6200 332640 695520 950976 Harshad 63 84 195 + 999852 heptagonal 3186 3744 6426 + 991305 hex 4219 32137 115837 + 841111 hexagonal 276 630 861 + 976503 highly composite 840 20160 45360 + 665280 hoax 84 438 476 + 999860 Hogben 381 553 993 + 995007 Honaker 1049 1571 2591 + 996011 house 933 1716 5336 + 654278 hungry 37929 hyperperfect 26977 iban 173 272 371 + 777771 iccanobiF 1053 idoneal 840 impolite 8192 524288 interprime 195 266 381 + 999993 Jacobsthal 87381 Jordan-Polya 216 432 1536 + 829440 junction 216 416 521 + 999938 Kaprekar 95121 142857 148149 + 961038 katadrome 62 63 65 + 987650 Lehmer 771 3855 8481 + 983055 Leyland 945 1649 20412 + 423393 lonely 15704 15705 24281 + 492170 Lucas 521 2207 103682 271443 lucky 63 75 195 + 999951 Lynch-Bell 216 432 612 + 984312 m-pointer 11261 22511 61211 magic 65 4641 21455 + 976625 magnanimous 65 443 536 + 931130 modest 266 411 422 + 999801 Moran 63 84 195 + 999036 Motzkin 310572 narcissistic 371 nialpdrome 62 63 65 + 999995 nonagonal 75 261 750 + 985536 nude 216 432 488 + 999936 O'Halloran 84 oban 63 65 75 + 993 octagonal 65 645 936 + 990725 odious 62 84 161 + 999995 Ormiston 1931 25031 35879 + 969131 palindromic 161 272 383 + 969969 palprime 383 16661 30703 + 73637 pancake 326 466 497 + 996167 pandigital 75 216 1646 + 800661 partition 26015 37338 53174 + 614154 pentagonal 1926 2147 3876 + 991047 pernicious 62 65 84 + 999995 Perrin 486 1497 1983 + 313007 Pierpont 65537 plaindrome 266 377 466 + 689999 Poulet 645 4371 8481 + 769757 power 216 441 1521 + 962361 powerful 216 432 441 + 996872 practical 84 216 272 + 999978 prim.abundant 272 426 438 + 999858 prime 173 383 431 + 999953 primeval 1037 10237 10367 + 103679 pronic 272 552 650 + 995006 Proth 65 161 993 + 994305 pseudoperfect 84 216 272 + 999980 rare 65 repfigit 75 31331 repunit 63 381 553 + 995007 Rhonda 4752 5265 8526 + 733125 Ruth-Aaron 492 493 1521 + 996558 Saint-Exupery 7500 16320 40260 + 937500 Sastry 106755 self 75 266 411 + 999978 self-describing 224444 semiprime 62 65 95 + 999995 sliding 65 650 785 + 785000 Smith 438 483 645 + 999536 sphenic 195 266 426 + 999971 square 441 1521 2601 + 962361 star 3037 16537 57037 + 683437 straight-line 432 531 543 + 98765 strobogrammatic 986 6119 9116 + 996966 strong prime 431 461 521 + 999953 super Niven 630 840 2016 + 936000 super-d 261 381 431 + 999931 superabundant 840 332640 665280 tau 84 276 372 + 999896 taxicab 4104 32832 39312 + 994688 tetrahedral 84 816 3654 + 924176 tetranacci 147312 547337 triangular 276 630 861 + 980700 tribonacci 927 223317 trimorphic 75 9375 59375 + 281249 truncatable prime 173 383 443 + 979337 twin 431 461 521 + 999331 uban 62 63 65 + 95 Ulam 62 431 441 + 999980 undulating 161 272 383 + 959595 unprimeable 326 516 536 + 999995 untouchable 216 276 326 + 999912 upside-down 1649 3287 3647 + 999111 vampire 102510 110758 135828 + 841995 wasteful 62 63 65 + 999995 weak prime 383 443 491 + 999931 weakly prime 584141 weird 9272 10430 16310 + 992390 Wieferich 94797 127881 473985 Woodall 63 383 4373 + 491519 Zeisel 59081 287979 Zuckerman 216 432 612 + 973728 Zumkeller 84 216 272 + 99996 zygodrome 1166 3311 4466 + 999933