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fibodiv numbers
These are numbers  $n$  whose representation can be split into two numbers, say  $a$  and  $b$, such that the Fibonacci-like sequence which uses  $a$  and  $b$  as seeds contains  $n$  itself.

For example, 549 is a fibodiv since it can be divided into 54 and 9 and they produce the sequence 54, 9, 63, 72, 135, 207, 342, 549,... containing 549 itself.

Note that the 6 fibodiv numbers with 2 digits are, by definition, repfigit numbers, too.

It is easy to see that there are infinite fibodiv numbers, since the numbers 19, 199, 1999,... and 28, 298, 2998,... are all fibodiv numbers.

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

741576505071555
676527025472856
689537545866351

The first fibodiv numbers are 14, 19, 28, 47, 61, 75, 122, 149, 183, 199, 244, 298, 305, 323, 366, 427, 488, 497, 549 more terms

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

a-pointer 3089 aban 14 19 28 47 + 969 abundant 366 3248 5466 6496 + 47880390 admirable 366 5466 7806 18534 + 44688364 alt.fact. 19 alternating 14 47 61 149 + 65454505 amenable 28 61 149 244 + 984380344 anti-perfect 244 apocalyptic 366 497 646 1301 + 29998 arithmetic 14 19 47 61 + 9869390 balanced p. 123047543 betrothed 75 binomial 28 969 brilliant 14 323 130901 609999623 c.decagonal 61 911 14311 c.nonagonal 28 c.square 61 1301 8845 c.triangular 19 199 1999 Carol 47 Catalan 14 Chen 19 47 149 199 + 67645819 congruent 14 28 47 61 + 9869390 Cunningham 28 122 244 323 + 17690 Curzon 14 366 13010 16913 + 145624910 cyclic 19 47 61 149 + 9166661 D-number 183 2733 3903 9267 + 4788039 d-powerful 2733 24712 29923 32525 + 209987 de Polignac 149 24719 40331 61147 + 30324247 deficient 14 19 47 61 + 9869390 dig.balanced 19 75 149 7995 + 199546494 Duffinian 75 183 305 323 + 8899945 eban 52040 economical 14 19 47 61 + 19999999 emirp 149 199 1301 1499 + 9161 emirpimes 122 183 497 2602 + 94164767 equidigital 14 19 47 61 + 19999999 esthetic 323 evil 75 149 183 298 + 974999994 Friedman 15612 54642 124896 frugal 240638283 721914849 gapful 13010 14311 15612 16913 + 11240023125 good prime 149 happy 19 28 2602 2998 + 8737446 harmonic 28 Harshad 9744 13010 23418 26020 + 9441619425 hex 19 61 29107 hexagonal 28 hoax 1822 6178 22117 27321 + 70224572 Hogben 183 87321 Honaker 1301 67645819 123047543 hyperperfect 28 iban 14 47 122 244 + 100177 idoneal 28 inconsummate 75 488 497 549 + 974994 interprime 969 4555 9267 11709 + 95760780 junction 305 911 11709 13010 + 87374946 katadrome 61 75 87321 Lucas 47 199 lucky 75 427 6505 9267 + 3769767 m-pointer 61 magic 66351 magnanimous 14 47 61 metadrome 14 19 28 47 + 12356 modest 19 199 911 1499 + 1999999999 Moran 23418 927267 38304312 43092351 Motzkin 323 nialpdrome 61 75 911 9744 + 98876 nonagonal 75 969 nude 122 244 366 488 + 114912936 oban 19 28 75 305 + 969 odious 14 19 28 47 + 986999390 palindromic 323 646 969 pancake 497 panconsummate 14 61 pandigital 19 75 32525 799995 1042765893 perfect 28 pernicious 14 19 28 47 + 9869390 Pierpont 19 plaindrome 14 19 28 47 + 1999999999 practical 28 3248 6496 9744 + 2329856 prim.abundant 366 5466 7806 18534 + 44688364 prime 19 47 61 149 + 8613332801 Proth 79361 pseudoperfect 28 366 3248 5466 + 974994 repfigit 14 19 28 47 + 75 repunit 183 87321 Ruth-Aaron 3248 Sastry 183 self 75 244 323 3089 + 984380344 semiprime 14 122 183 298 + 97356793 Smith 1822 6178 22117 27321 + 86184702 sphenic 366 646 795 969 + 98952806 strong prime 149 1301 2087 199999 + 21890647 super-d 19 969 3379 6377 + 6384052 tau 488 7288 10408 12992 + 984380344 tetrahedral 969 triangular 28 tribonacci 149 trimorphic 75 truncatable prime 47 twin 19 61 149 199 + 2087 uban 19 28 47 61 75 Ulam 28 47 497 646 + 2472712 undulating 323 646 969 unprimeable 3248 4555 5466 6505 + 9749994 untouchable 1822 2602 5204 6178 + 974994 upside-down 19 28 wasteful 28 75 244 298 + 9869390 weak prime 19 47 61 199 + 67645819 weird 91070 Woodall 323 Zumkeller 28 366 3248 5466 + 97494 zygodrome 114488