Self-describing number

A self-describing number is a natural number m, in which the nth digit indicates the frequency with which the digit n-1 occurs in this number. The number is thus self-describing in the sense that the number can be reconstructed solely from knowing its digits. It is b places long, consists only of the digits 0, 1,…, b-1 and is specified in the base b .

Examples

An example of a number that is self-describing in the decimal (base 10) system is 6210001000, since the number contains six zeros, two ones, one two, zero three, zero fours, zero five, one six, zero sevens, zero eight, and zero nines .

Some self-describing figures can be found in the following table:

Base b	self-describing number in this base b (sequence A138480 in OEIS )	Value of the number in the decimal system (sequence A108551 in OEIS )
${\displaystyle 4}$	${\displaystyle 1210,2020}$	${\displaystyle {\underline {1}}\cdot 4^{3}+{\underline {2}}\cdot 4^{2}+{\underline {1}}\cdot 4^{1}+{\underline {0}}\cdot 4^{0}=64+32+4=100}$ ${\displaystyle {\underline {2}}\cdot 4^{3}+{\underline {0}}\cdot 4^{2}+{\underline {2}}\cdot 4^{1}+{\underline {0}}\cdot 4^{0}=2\cdot 64+2\cdot 4=136}$
${\displaystyle 5}$	${\displaystyle 21200}$	${\displaystyle {\underline {2}}\cdot 5^{4}+{\underline {1}}\cdot 5^{3}+{\underline {2}}\cdot 5^{2}=2\cdot 625+125+2\cdot 25=1425}$
${\displaystyle 7}$	${\displaystyle 3211000}$	${\displaystyle {\underline {3}}\cdot 7^{6}+{\underline {2}}\cdot 7^{5}+{\underline {1}}\cdot 7^{4}+{\underline {1}}\cdot 7^{3}=3\cdot 117649+2\cdot 16807+2401+343=389305}$
${\displaystyle 8}$	${\displaystyle 42101000}$	${\displaystyle {\underline {4}}\cdot 8^{7}+{\underline {2}}\cdot 8^{6}+{\underline {1}}\cdot 8^{5}+{\underline {1}}\cdot 8^{3}=4\cdot 2097152+2\cdot 262144+32768+512=8946176}$
${\displaystyle 9}$	${\displaystyle 521001000}$	${\displaystyle {\underline {5}}\cdot 9^{8}+{\underline {2}}\cdot 9^{7}+{\underline {1}}\cdot 9^{6}+{\underline {1}}\cdot 9^{3}=5\cdot 43046721+2\cdot 4782969+531441+729=225331713}$
${\displaystyle 10}$	${\displaystyle 6210001000}$	${\displaystyle {\underline {6}}\cdot 10^{9}+{\underline {2}}\cdot 10^{8}+{\underline {1}}\cdot 10^{7}+{\underline {1}}\cdot 10^{3}=6\cdot 1000000000+2\cdot 100000000+10000000+1000=6210001000}$
${\displaystyle 11}$	${\displaystyle 72100001000}$	${\displaystyle {\underline {7}}\cdot 11^{10}+{\underline {2}}\cdot 11^{9}+{\underline {1}}\cdot 11^{8}+{\underline {1}}\cdot 11^{3}=7\cdot 25937424601+2\cdot 2357947691+214358881+1331=186492227801}$
${\displaystyle 12}$	${\displaystyle 821000001000}$	${\displaystyle {\underline {8}}\cdot 12^{11}+{\underline {2}}\cdot 12^{10}+{\underline {1}}\cdot 12^{9}+{\underline {1}}\cdot 12^{3}=6073061476032}$
${\displaystyle 13}$	${\displaystyle 9210000001000}$	${\displaystyle {\underline {9}}\cdot 13^{12}+{\underline {2}}\cdot 13^{11}+{\underline {1}}\cdot 13^{10}+{\underline {1}}\cdot 13^{3}=213404945384449}$
${\displaystyle 14}$	${\displaystyle A2100000001000}$	${\displaystyle {\underline {10}}\cdot 14^{13}+{\underline {2}}\cdot 14^{12}+{\underline {1}}\cdot 14^{11}+{\underline {1}}\cdot 14^{3}=8054585122464440}$
${\displaystyle 15}$	${\displaystyle B21000000001000}$	${\displaystyle {\underline {11}}\cdot 15^{14}+{\underline {2}}\cdot 15^{13}+{\underline {1}}\cdot 15^{12}+{\underline {1}}\cdot 15^{3}=325144322753909625}$
${\displaystyle 16}$	${\displaystyle C210000000001000}$	${\displaystyle {\underline {12}}\cdot 16^{15}+{\underline {2}}\cdot 16^{14}+{\underline {1}}\cdot 16^{13}+{\underline {1}}\cdot 16^{3}=13983676842985394176}$
${\displaystyle \ldots }$	${\displaystyle \ldots }$	${\displaystyle \ldots }$

With bases greater than 10, it is common to use A = 10, B = 11, C = 12, etc. for lack of further digits.

properties

• In the case of self-describing numbers, the number of digits in the number is equal to the base b (according to the definition).
• In the case of self-describing numbers, the sum of the digits is equal to the number of digits in the number.
• For self-describing numbers, the sum of the digits is equal to base b .
• A self-describing number is always a multiple of its base b .
• In the case of self-describing numbers, the last digit (in the ones place , which indicates how often the digit b-1 occurs in the number, written in the base b ) is always a zero.
• Self - describing numbers in base b are always Harshad numbers (that is, they are always divisible by their digit sum when they are written in base b ).
• There are no self-describing numbers made up of two, three or six digits.

generalization

If you allow the number of digits to be smaller than the base b , but the digits still indicate how often they appear in the number, the number is called an autobiographical number .

Example: The number 42101000 is in the base 8 (with the eight digits 0 to 7) a self-describing number because it consists of 4 zeros, 2 ones, 1 twos, 0 threes, 1 fours, 0 fives, 0 sixes and 0 sevens . There are no more digits in the eight-digit system . The number 42101000 is not a self-describing number in the decimal system (i.e. with the base 10 with the ten digits 0 to 9) because it does not have 10 digits. But it is an autobiographical number because, as before, it still consists of 4 zeros, 2 ones, 1 twos, 0 threes, 1 fours, 0 fives, 0 sixes and 0 sevens. There are more digits in the decimal system (the number of eights and nines is missing in the number), but with autobiographical numbers it is not necessary to state all higher digits as long as they do not appear in the number.

literature

• Clifford Pickover: Keys to Infinity . Wiley, New York 1995, ISBN 978-0-471-19334-0 , Chapter 28: “Chaos in Ontario”, pp. 217-219,

Web links

• Autobiographical Numbers. UVa Online Judge, accessed on May 26, 2018 . 
• Eric W. Weisstein : Self-descriptive number . In: MathWorld (English).
• Self- describing numbers up to 10 digits: Sequence A046043 in OEIS