GOTW- Counting - Page 20

• 148 is the second number to be both a heptagonal number and a centered heptagonal number (the first is 1)
• 148 is the twelfth member of the Mian–Chowla sequence
• There are 148 perfect graphs with 6 vertices

• 149 is the 35th prime number, and with the next prime number, 151, is a twin prime, thus 149 is a Chen prime. 149 is a strong prime in the sense that it is more than the arithmetic mean of its two neighboring primes. 149 is an Eisenstein prime with no imaginary part and a real part of the form .

• Given 149, the Mertens function returns 0. 149 is a tribonacci number, being the sum of the three preceding terms, 24, 44, 81.

• 149 is a strictly non-palindromic number, meaning that it is not palindromic in any base from binary to base 147. However, in base 10, it is a full reptend prime, since the decimal expansion of 1/149 repeats 0067114093959731543624161073825503355704697986577181208053691275167785234899328859060402684563758389261744966442953020134228187919463087248322147651 infinitely.

• 50 is the sum of eight consecutive primes (7 + 11 + 13 + 17 + 19 + 23 + 29 + 31). Given 150, the Mertens function returns 0.

• The sum of Euler's totient function f(x) over the first twenty-two integers is 150.

• 150 is a Harshad number and an abundant number.

151 is the 36th prime number, the previous is 149, with which it comprises a twin prime. 151 is also a palindromic prime. 151 is a centered decagonal number. 151 is also a lucky number.

151 appears in the Padovan sequence, preceded by the terms 65, 86, 114 (it is the sum of the first two of these).

152 is the sum of four consecutive primes (31 + 37 + 41 + 43). It is a nontotient since there is no integer with 152 coprimes below it.

152 is a refactorable number since it is divisible by the total number of divisors it has, and in base 10 it is divisible by the sum of its digits, making it a Harshad number

As a triangular number, 153 is the sum of the first 17 integers, and it also the sum of the first five positive factorials:.^[1]

The number 153 is also a hexagonal number, and a truncated triangle number, meaning that 1, 15, and 153 are all triangle numbers.

The distinct prime factors of 153 add up to 20, and so do the ones of 154, hence the two form a Ruth-Aaron pair.

Since , it is a 3-narcissistic number, and it is also the smallest three-digit number which can be expressed as the sum of cubes of its digits.^[2] Only five other numbers can be expressed as the sum of the cubes of their digits: 0, 1, 370, 371 and 407.^[3] It is also a Friedman number, since 153 = 3 51, and a Harshad number in base 10, being divisible by the sum of its own digits.

The Biggs'Smith graph is a symmetric graph with 153 edges, all equivalent.

Another interesting feature of the number 153 is that it is the limit of the following algorithm:^[4][5]

1. Take a random positive integer, divisible by three.
2. Split that number into its base 10 digits.
3. Take the sum of their cubes.
4. Go back to the second step.

An example, starting with the number 84:

• 154 is a nonagonal number. Its factorization makes 154 a sphenic number
• There is no integer with exactly 154 coprimes below it, making 154 a noncototient, nor is there, in base 10, any integer that added up to its own digits yields 154, making 154 a self number
• 154 is the sum of the first six factorials
• With just 17 cuts, a pancake can be cut up into 154 pieces (Lazy caterer's sequence)
• Distinct prime factors of 154 add up to 20, and so do the ones of 153, hence the two form a Ruth-Aaron pair
• 154! + 1 is a factorial prime

155 (one hundred fifty-five or one hundred and fifty-five) is the natural number following 154 and preceding 156. It is a composite number.

If you add up all the primes between the least and greatest prime factors of 155, that is, 5 and 31, the result is 155. (sequence A055233 in OEIS)

There are 155 primitive permutations of degree 81. ?A000019

It is a pronic number, a dodecagonal number, a refactorable number and a Harshad number.

156 is a repdigit in base 5 (1111).