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insolite numbers
A number  $n$  is said to be insolite if it is divisible by the sum and by the product of the squares of its digits.

For example, 122121216 is insolite since it is divisible by

\[1^2+2^2+2^2+1^2+2^2+1^2+2^2+1^2+6^2=56\]
and by
\[1^2\cdot2^2\cdot2^2\cdot1^2\cdot2^2\cdot1^2\cdot2^2\cdot1^2\cdot6^2=9216.\]

There are only 428 insolite numbers below  $10^{20}$.

Only few insolite numbers are known to include the digit '5'. Indeed, the zerolessness conditions prescribe that in this case the number is made of an odd number of odd digits.

The table below reports the smallest insolite numbers containing a specific digit.

dn
2 11112, 1122112, 122121216, 1111112112
3 1111211136, 11111113116, 11132111232
4 11121114112, 121121114112, 211121114112
5 1111111111131111131111111111175
6 122121216, 1111211136, 1116122112
7 123412112474112, 211912113131712
8 121111216128, 911131213824
9 911131213824, 211912113131712

The second smallest insolite number cointaining the digit 5 could by

\[1111111111111111117111111111911111375,\]
while the smallest insolite number containing all the digits except 5 is 711813411914121216.

Every number greater than 2644145 can be written as the sum of insolite numbers.

The first insolite numbers are 111, 11112, 1122112, 111111111, 122121216, 1111112112, 1111211136, 1116122112, 1211162112, 11111113116 more terms

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

A graph displaying how many insolite numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of insolite numbers
Useful links OEIS, Sequence A098034
Insolite numbers can also be... (you may click on names or numbers and on + to get more values)

aban 111 abundant 11112 1122112 amenable 11112 1122112 122121216 apocalyptic 11112 arithmetic 111 11112 congruent 111 1122112 D-number 111 deficient 111 dig.balanced 11112 Duffinian 111 economical 111 1122112 equidigital 111 1122112 evil 111 1122112 111111111 gapful 11112 122121216 1111112112 1111211136 1116122112 1211162112 11111121216 11112122112 11132111232 11133122112 + 12111213312 21111311232 31111221312 32111111232 Harshad 111 11112 111111111 122121216 1111112112 1111211136 1116122112 1211162112 Hogben 111 iban 111 11112 interprime 111 junction 111 lucky 111 magic 111 modest 111 111111111 Moran 111 nialpdrome 111 111111111 nonagonal 111 nude 111 11112 1122112 111111111 122121216 odious 11112 122121216 palindromic 111 111111111 pernicious 11112 plaindrome 111 11112 111111111 practical 1122112 pseudoperfect 11112 repdigit 111 111111111 repunit 111 111111111 self 11112 semiprime 111 straight-line 111 111111111 strobogrammatic 111 111111111 wasteful 11112 Zuckerman 111 11112 1122112 111111111 122121216 1111112112 1111211136 1116122112 1211162112 Zumkeller 11112 zygodrome 111 111111111