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Homeself-descriptive number

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# self-descriptive number

A self-descriptive number $n$ in base $b$ is an integer such that each base $b$ digit

$d_{x}=\sum_{{d_{i}=x}}1$ |

where each $d_{i}$ is a digit of $n$, $i$ is a very simple, standard iterator operating in the range $-1<i<b$, and $x$ is a position of a digit; thus $n$ “describes” itself.

For example, the integer 6210001000 written in base 10. It has six instances of the digit 0, two instances of the digit 1, a single instance of the digit 2, a single instance of the digit 6 and no instances of any other base 10 digits.

Base 4 might be the only base with two self-descriptive numbers, $1210_{4}$ and $2020_{4}$. From base 7 onwards, every base $b$ has at least one self-descriptive number of the form $(b-4)^{{b-1}}+2b^{{b-2}}+b^{{b-3}}+b^{4}$. It has been proven that 6210001000 is the only self-descriptive number in base 10, but it’s not known if any higher bases have any self-descriptive numbers of any other form.

Sequence A108551 of the OEIS lists self-descriptive numbers from quartal to hexadecimal.

## Mathematics Subject Classification

11A63*no label found*

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## Corrections

Summation? by CWoo ✓

what does d_x mean? by CWoo ✓

Definition before example by CompositeFan ✓

## Comments

## how to describe algebraically

I'm not entirely sure d_x = \sum_{d_i = x} is correct. I understand what it means (digit y at position x is the total number of instances of digit y in the number) but I'm not sure if the formula is mathematically correct. Can this be fed to a CAS so that it understands what is meant?