GOTW- Counting - Page 18

One hundred [and] twenty-eight is the seventh power of 2. It is the largest number which cannot be expressed as the sum of any number of distinctsquares.^[1][2] But it is divisible by the total number of its divisors, making it a refactorable number.^[3]

The sum of Euler's totient function f(x) over the first twenty integers is 128.^[4]

128 can be expressed by a combination of its digits with mathematical operators thus 128 = 2^{8 - 1}, making it a Friedman number in base 10.^[5]

A hepteract has 128 vertices.

128 is the only 3-digit number that is a 7th power. 2 to the 7th power.

129 is the sum of the first ten prime numbers. It is the smallest number that can be expressed as a sum of three squares in four different ways: , , , and .

129 is the product of only two primes, 3 and 43, making 129 a semiprime. Since 3 and 43 are both Gaussian primes, this means that 129 is a Blum integer.

129 is a repdigit in base 6 (333).

129 is a happy number.

130 is a sphenic number. It is a noncototient since there is no answer to the equation x - f(x) = 130.

130 is the only integer that is the sum of the squares of its first four divisors, including 1: 1² + 2² + 5² + 10² = 130.

130 is the largest number that cannot be written as the sum of four hexagonal numbers.^[1]

131 is a Sophie Germain prime, the second 3-digit palindromic prime, and also a permutable prime with 113 and 311. It can be expressed as the sum of three consecutive primes, 131 = 41 + 43 + 47. 131 is an Eisenstein prime with no imaginary part and real part of the form . Because the next odd number, 133, is a semiprime, 131 is a Chen prime.

131 is a full reptend prime in base 10. The decimal expansion of 1/131 repeats the digits 0076335877862595419847328244274809160305343511450381679389312977099236641221374045801526717557251908396946564885496183206106870229 indefinitely.

132 is the sixth Catalan number. It is a pronic number, the product of 11 and 12. As it has 12 divisors total, 132 is a refactorable number.

If you take the sum of all 2-digit numbers you can make from 132, you get 132: . 132 is the smallest number with this property.^[1]

But there is no number that, when added to the sum of its own digits, adds up to 132, making 132 a self number.However, 132 is a Harshad number, divisible by the sum of its base 10 digits.

133 is an n whose divisors (excluding n itself) added up divide f(n). It is an octagonal number and a Harshad number. It is also a happy number.

133 is a repdigit in base 11 (111), whilst in base 20 it is a cyclic number formed from the reciprocal of the number three.

133 is a semiprime: a product of two prime numbers, namely 7 and 19. Since those prime factors are Gaussian primes, this means that 133 is a Blum integer.

Two consecutive Harshad Numbers 😆

Sheils you missed it 🤣

134 is a nontotient since there is no integer with exactly 134 coprimes below it. And it is a noncototient since there is no integer with 134 integers with common factors below it. 134 is .

In Roman numerals, 134 is a Friedman number since CXXXIV = XV * (XC/X) - I.

This number in base 10 can be expressed in operations using its own digits in at least two different ways. One is as a sum-product number,

(1 and 144 share this property) and the other is as the sum of consecutive powers of its digits:

(175, 518, and 598 also have this property).

135 is a Harshad number.😎

There are a total of 135 primes between 1,000 and 2,000.

for . This polynomial plays an essential role in Apry's proof that is irrational.

One hundred [and] thirty-seven is the 33rd prime number; the next is 139, with which it comprises a twin prime, and thus 137 is a Chen prime. 137 is an Eisenstein prime with no imaginary part and a real part of the form . It is also the fourth Stern prime. 137 is a strong prime in the sense that it is more than the arithmetic mean of its two neighboring primes.

Using two radii to divide a circle according to the golden ratio yields sectors of approximately 137 (the golden angle) and 222.

137 is a strictly non-palindromic number and a primeval number.

The fifth harmonic number is