GOTW- Counting - Page 17

• 123 is a Lucas number.^[1] It is the eleventh member of the Mian-Chowla sequence

124 is the sum of eight consecutive primes (5 + 7 + 11 + 13 + 17 + 19 + 23 + 29). It is a nontotient since there is no integer with 124 coprimes below it, it is an untouchable number since there is no integer whose proper divisorsadd up to 124.

In base 5 it is a repdigit.

One hundred [and] twenty-five is the cube of 5. It can be expressed as a sum of two squares in two different ways, 125 = 10 + 5 = 11 + 2.^[1]

125 and 126 form a Ruth-Aaron pair under the second definition in which repeated prime factors are counted as often as they occur.

Like so many other powers of 5, it is a Friedman number in base 10 since 125 = 5^{1 + 2}.

One hundred [and] twenty-six is a pentagonal pyramidal number, and a decagonal number as well as a pentatope number.

It is an abundant number. However, it is the fifth Granville number, and the third such not to be a perfect number. Also, it is known to be the smallest Granville number with three distinct prime factors, and perhaps the only such Granville number.^[1]

125 and 126 form a Ruth-Aaron pair under the second definition in which repeated prime factors are counted as often as they occur.

In base 10, it is a Harshad number as well as a Friedman number, since 126 = 6 21.😃

On the standard 3x3 Rubik's Cube, starting from the original set position, the basic algorithm of alternating top rotation- side rotation will return the cube to its original state in 126 moves.