| Number: | 138 |
| Divisor: | 1, 2, 3, 6, 23, 46, 69, 138 |
138 (one hundred [and] thirty-eight) is the natural number following 137 and preceding 139.
138 is a sphenic number,[1] and the smallest product of three primes such that in base 10, the third prime is a concatenation of the other two:
2 ⋅ 3 ⋅ 23
138 has eight total divisors that generate an arithmetic mean of 36,[2] which is the eighth triangular number.[3] While the sum of the digits of 138 is 12, the product of its digits is 24.[4]
138 is an Ulam number,[5] the thirty-first abundant number,[6] and a primitive (square-free) congruent number.[7] It is the first non-trivial 47-gonal number.[8]
As an interprime, 138 lies between the eleventh pair of twin primes (137, 139),[9] respectively the 33rd and 34th prime numbers.[10]
It is the sum of two consecutive primes (67 + 71),[11] and the sum of four consecutive primes (29 + 31 + 37 + 41).
There are a total of 44 numbers that are relatively prime with 138 (and up to),[12] while 22 is its reduced totient.[13]
Bn
A magic sum of 138 is generated inside four magic circles that features the first thirty-three non-zero integers, with a 9 in the center (first constructed by Yang Hui).
The simplest Catalan solid, the triakis tetrahedron, produces 138 stellations (depending on rules chosen), 44 of which are fully symmetric and 94 of which are enantiomorphs.[16]
Using two radii to divide a circle according to the golden ratio yields sectors of approximately 138 degrees (the golden angle), and 222 degrees.