WON TOPIC 83

World!Of
Numbers

WON plate
83 |

Repdigits, or numbers with repeated digits

For repdigit 22 see WONplate 32

For repdigit 55 see WONplate 81

The repdigit 77 can be written as the sum of three squared consecutives
4² + 5² + 6²
See also WONplate 43

The repdigit 88 can be written as the sum of four consecutive primes
17 + 19 + 23 + 29
See also WONplate 32

The repdigit 222 can be expressed as the sum of just two consecutive primes
109 + 113

The repdigit 1111 can be the result of various transformations of the prime 11
1. Concatenate 11 with itself.
2. Multiply 11 by 101.
3. ...
But the most extraordinary way must surely be this one :

11² + 12² + 13² + 14² + 15² + 16² = 1111

In borderprime world 1111 occurs a few time as displacement value

10⁷³¹ + 1111
10¹⁵¹² + 1111
10¹⁷⁸⁷ + 1111
10²¹⁴⁶ + 1111
10³⁹⁸⁷ + 1111
10⁴⁰⁵⁶ + 1111
10⁵²⁹⁶ + 1111

So far only these 7 positive displacements have been discovered !

The repdigit 8888 can be expressed as the sum of just two consecutive primes
4441 + 4447

The following repdigits are, once converted into hexadecimal base, also palindromic !
333333 = 51615
666666 = A2C2A

The repdigit number 777777 can be expressed as the sum of three consecutive primes
259229 + 259271 + 259277

Two repdigit numbers of the form (4)_n can be expressed as the sum
of the squares of consecutive numbers (#8 and #63 resp.)
n = 5 44444 71² + 72² + ... + 77² + 78²
n = 6 444444 51² + 52² + ... + 112² + 113²

A010785 Repdigit numbers, or numbers with repeated digits. - N. J. A. Sloane
A028987 Repdigit - 1 is prime (next term is 88...8 (72 digits)). - Patrick De Geest
A028988 Repdigit + 1 is prime (next term is 22...2 (36 digits)). - Patrick De Geest

A000083

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