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upside-down numbers
R.E.Kennedy & C.N.Cooper called a number  $n$  upside-down number if its i-th leftmost digit and its i-th rightmost digit are complements, i.e., their sum is 10.

For example,  $42586$  is an upside-down number because  $4+6=2+8=5+5=10$.

From the definition it follows that upside-down numbers are zeroless and if they have an odd number of digits, then the central digit is 5.

Kennedy and Cooper proved that for the  $n$-th upside-down number  $u_n$  it holds

\[
\left(\frac{4n}{15}\right)^{\frac{\log 10}{\log 3}} \le u_n \le\left(\frac{81n}{16}\right)^{\frac{\log 10}{\log 3}}\,.
\]

The first upside-down numbers are 5, 19, 28, 37, 46, 55, 64, 73, 82, 91, 159, 258, 357, 456, 555, 654, 753, 852, 951, 1199, 1289, 1379 more terms

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

a-pointer 11946199 12991189 19537519 + 9971739311 ABA 64 2738 8192 151959 aban 19 28 37 + 951 abundant 258 456 654 + 49991116 Achilles 22128988 22817137939288 admirable 258 654 2828 + 89491612 alt.fact. 19 alternating 258 456 654 + 989852121 amenable 28 37 64 + 989951121 apocalyptic 1199 1289 1739 + 29518 arithmetic 19 37 46 + 9995111 balanced p. 7823 92581 3355577 + 9882648221 binomial 28 55 91 + 485628284526 brilliant 1649 1739 1829 + 97691431 c.decagonal 9461 c.heptagonal 3175397 3695147 96119941 + 77519519533 c.nonagonal 28 55 91 c.octagonal 926654481 c.pentagonal 456 951 95551 43482676 c.square 7928552813 98941355796121 c.triangular 19 46 64 cake 64 Chen 19 37 1289 + 99791311 congruent 28 37 46 + 9995111 constructible 64 8192 cube 64 Cunningham 28 37 82 + 241142869968 Curzon 1289 1469 1649 + 982821 cyclic 19 37 73 + 9985211 D-number 159 753 951 + 994611 d-powerful 357 2738 4736 + 9785231 de Polignac 1199 1649 1829 + 99973111 deceptive 91 16491649 37373737 deficient 19 37 46 + 9995111 dig.balanced 19 37 555 + 199852119 Duffinian 55 64 1379 + 9995111 eban 46 64 economical 19 37 64 + 19982119 emirp 37 73 1559 + 99864211 emirpimes 159 951 1379 + 99928111 equidigital 19 37 64 + 19982119 eRAP 59713579315 esthetic 456 654 34567 + 898987654321212 evil 46 159 258 + 999654111 fibodiv 19 28 Fibonacci 55 Friedman 8192 183729 261948 + 786423 frugal 8192 144669 151959 + 992357811 gapful 1919 2828 3737 + 89976543112 good prime 37 9371 1245689 + 98773321 happy 19 28 82 + 9955511 harmonic 28 Harshad 555 53575 58525 + 599951115 heptagonal 55 13579 74563 71628493 hex 19 37 91 hexagonal 28 91 485628284526 hoax 456 654 45556 + 99682411 Hogben 73 91 98392919181721 Honaker 7283 11473699 13191979 + 98746321 hyperperfect 28 iban 73 3377 7373 idoneal 28 37 357 impolite 64 8192 inconsummate 1649 3287 3647 + 999111 interprime 64 3197 4286 + 99973111 Jordan-Polya 64 8192 junction 3827 4826 5825 + 96991141 Kaprekar 55 55555555555 katadrome 64 73 82 + 987654321 Lehmer 91 72583 12346789 99991111 Leyland 1649 lucky 37 73 159 + 997311 Lynch-Bell 243768 623784 836472 magic 1379 metadrome 19 28 37 + 123456789 modest 19 46 555 + 1919191919 Moran 555 53575 58525 5565455 nialpdrome 55 64 73 + 999999951111111 nonagonal 46 158259 171939 + 698214 nude 55 555 2288 + 55555555 oban 19 28 37 + 753 octagonal 4375376 5867463425 542458256865 7792795138133 odious 19 28 37 + 999951111 Ormiston 13146979 13282879 13373779 + 1978462319 palindromic 55 555 5555 + 555555555555555 pancake 37 46 1379 + 87948564526132 panconsummate 37 73 91 pandigital 19 2558 124689 + 282951828 pentagonal 852 5735 863742 346369147467 perfect 28 pernicious 19 28 37 + 9995111 Pierpont 19 37 73 plaindrome 19 28 37 + 555555555555555 power 64 8192 45346756 + 43211599876 powerful 64 8192 22128988 + 22817137939288 practical 28 64 456 + 8765432 prim.abundant 258 654 2828 + 89491612 prime 19 37 73 + 99998521111 primeval 37 1379 pronic 4556 8372 849162 + 886422 Proth 7553 14573569 162658254849 795855552513 pseudoperfect 28 258 456 + 899112 repdigit 55 555 5555 + 555555555555555 repfigit 19 28 repunit 73 91 41182996 98392919181721 Ruth-Aaron 19191919 self 64 951 1199 + 699951114 semiprime 46 55 82 + 99928111 Smith 654 852 5915 + 99682411 sphenic 258 357 555 + 99991111 square 64 45346756 68591524 + 43211599876 star 37 73 straight-line 159 258 357 + 555555555555555 strong prime 37 1289 3467 + 99919111 super-d 19 753 1469 + 9955511 tau 852 2648 4376 + 899357112 triangular 28 55 91 + 485628284526 truncatable prime 37 73 3467 + 7823 twin 19 73 1289 + 99955111 uban 19 28 37 + 91 Ulam 28 82 258 + 9945611 undulating 1919 2828 3737 + 91919191919191 unprimeable 2198 2828 4196 + 8965412 untouchable 852 2198 2828 + 899112 vampire 23428678 71673493 wasteful 28 46 55 + 9995111 weak prime 19 73 1559 + 99955111 Woodall 159 Zumkeller 28 258 456 + 81592 zygodrome 55 555 1199 + 999999555111111