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A number that it is divisible by the sum and by the product of the squares of its digits. more

The insolite numbers up to 1015 :
111, 11112, 1122112, 111111111, 122121216, 1111112112, 1111211136, 1116122112, 1211162112, 11111113116, 11111121216, 11112122112, 11121114112, 11132111232, 11133122112, 11213111232, 11311322112, 12111213312, 21111311232, 31111221312, 32111111232, 111122111232, 111132122112, 111211322112, 111312122112, 112111322112, 113112122112, 121111216128, 121111322112, 121121114112, 131111132112, 131112122112, 211111322112, 211121114112, 311112122112, 911131213824, 1111111113312, 1111121114112, 1121313321216, 1331611322112, 11111111114112, 11111111211312, 11111111312112, 11126112141312, 11211111111312, 11221121114112, 11311111112112, 11611142111232, 12111212122112, 12111213146112, 12113411162112, 12121141211136, 12142111113216, 21111212122112, 21132161114112, 21214111113216, 21311114121216, 41111131226112, 111111111111312, 111111111122112, 111111311111112, 112221411213312, 112264112111616, 123111311118336, 123412112474112, 211211261116416, 211912113131712, 311111111111112, 311111111411136, 323113121114112, 691112321114112.

Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 71 values, from 111 to 691112321114112).

n\r 0  1 
2692 2 
36722 3 
469002 4 
50155411 5 
66502220 6 
71869910613 7 
8680001002 8 
96212300210 9 
10025401013010 10 
1113413512236382

A pictorial representation of the table above
motab
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.