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

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

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

Useful links R.E.Kennedy & C.N.Cooper, Bach, 5465, and Upside-Down numbers,

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 productive 28 82 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 Sophie Germain 1289 1559 7193 + 9993557111 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