GOTW- Counting - Page 19

• Its factorization makes 138 a sphenic number
• The sum of four consecutive primes (29 + 31 + 37 + 41)
• The smallest product of 3 primes, such that in base 10, the third prime is a concatenation of the other two
• The third 47-gonal number
• An Ulam number (Sloane's A002858)
• Bond number 138
• 1 step palindrome

One hundred [and] thirty-nine is the 34th prime number, so it is divisible only by itself and 1. It is a twin prime with 137. Because 141 is asemiprime, 139 is a Chen prime. 139 is the smallest prime before a prime gap of length 10.^[1]

This number is the sum of five consecutive prime numbers (19 + 23 + 29 + 31 + 37).

It is the smallest factor of 64079, the smallest composite Lucas number with a prime index. It is also the smallest factor of the first nine terms of the Euclid–Mullin sequence, making it the tenth term.

139 is a strictly non-palindromic number.

140 is an abundant number and a harmonic divisor number. It is the sum of the squares of the first seven integers, which makes it a square pyramidal number, and in base 10 it is divisible by the sum of its digits, which makes it a Harshad number and a sheila number. 🤣

The sum of Euler's totient function f(x) over the first twenty-one integers is 140. An odious number because it has an oddnumber of ones in its binary number.

There is no answer to the equation f(x) = 142, making 142 a nontotient.

But there are 142 planar graphs with 6 unlabeled vertices.^[1]

143 is the sum of three consecutive primes (43 + 47 + 53), as well as the sum of seven consecutive primes (11 + 13 + 17 + 19 + 23 + 29 + 31). But this number is never the sum of an integer and its base 10 digits, making it a self number.

Every positive integer is the sum of at most 143 seventh powers (see Waring's problem).

143 is the difference in the first exception to the pattern shown below:

It is the twelfth Fibonacci number, and the largest one to also be a square,^[1] as the square of 12 (which is also its index in the Fibonacci sequence), following 89 and preceding 233.

144 is the smallest number with exactly 15 divisors.

144 is a number that is divisible by the value of its f function, which returns 48 in this case. Also, there are 21 solutions to the equation f(x) = 144, more than any integer below 144, making it a highly totient number.

, the smallest number whose fifth power is a sum of four (smaller) fifth powers. This solution was found in 1966 by L. J. Lander and T. R. Parkin, and disproved Euler's sum of powers conjecture.

The maximum determinant in a 9 by 9 matrix of zeroes and ones is 144.

144 is in base 10 a sum-product number, as well as a Harshad number

• Although composite, 145 is a pseudoprime.

• Given 145, the Mertens function returns 0.

• 145 is a pentagonal number and a centered square number.

• . 145 is the fourth number that is the sum of two different pairs of squares. Also, 145 is the result of 3⁴ + 4³, making it a Leyland number.

• , making it a factorion. The only other numbers that have the property that they are the sum of the factorials of their digits are 1, 2 and 40585.

146 is an octahedral number as well as a composite number.

It is a nontotient since there is no integer with 146 coprimes below it, and an untouchable number since there is no integer whose proper divisors add up to 146.

146 is a repdigit in base 8 (222).

There are 147 one-sided 6-polyhexes.

The digits forming 147 also form the left-hand column of a normal decimal numeric keypad.

The binary form of 147 is one of the binary numbers that contain all two-digit binary numbers (00, 01, 10 and 11) in a sequence.