138 (number)

← 137 138 139 →
← 130 131 132 133 134 135 136 137 138 139 → List of numbers Integers ← 0 100 200 300 400 500 600 700 800 900 →
Cardinal	one hundred thirty-eight
Ordinal	138th (one hundred thirty-eighth)
Factorization	2 × 3 × 23
Divisors	1, 2, 3, 6, 23, 46, 69, 138
Greek numeral	ΡΛΗ´
Roman numeral	CXXXVIII
Binary	100010102
Ternary	120103
Senary	3506
Octal	2128
Duodecimal	B612
Hexadecimal	8A16

138 (one hundred [and] thirty-eight) is the natural number following 137 and preceding 139.

In mathematics

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: ${\displaystyle 2\cdot 3\cdot 23}$. ^{[lower-alpha 1]} It is also a one-step palindrome in decimal (138 + 831 = 969).

Four concentric magic circles, with a magic constant of 138.

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 third 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). ^[12]

There are a total of 44 numbers that are relatively prime with 138 (and up to), ^[13] while 22 is its reduced totient. ^[14]

138 is the denominator of the twenty-second Bernoulli number ${\displaystyle B_{n}}$ (whose respective numerator, is 854513). ^{[15] [16]}

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). ^{[lower-alpha 2]}

The simplest Catalan solid, the triakis tetrahedron, produces 138 stellations (depending on rules chosen), ^{[lower-alpha 3]} 44 of which are fully symmetric and 94 of which are enantiomorphs. ^[17]

Using two radii to divide a circle according to the golden ratio yields sectors of approximately 138 degrees (the golden angle), and 222 degrees.

In science

• The Saros number of the solar eclipse series which began on June 6, 1472, and will end on July 11, 2716. The duration of Saros series 138 is 1244 years, and it contains 70 solar eclipses
• 138 Tolosa is a brightly colored, stony main belt asteroid
• The New General Catalogue object NGC 138, a spiral galaxy in the constellation Pisces
• 138P/Shoemaker-Levy is a periodic comet in the Solar System

In media

• "We Are 138", a 1978 song by the American punk rock band Misfits.
• Who's Afraid of 138!? is a trance record label operated by Dutch DJ Armin van Buuren, as a reference to the use of 138 BPM in some forms of trance music. ^{[18] [19]}

See also

• The year AD 138 or 138 BC
• List of highways numbered 138

Notes

1. The only other number less than 1000 in decimal with this property is 777 = 3 × 7 × 37.
2. This magic sum is generated from points that simultaneously lie on circles and diagonals, and, importantly, without including 9 in this sum (i.e. by bypassing it).
3. Using a different set of rules than Miller's rules for stellating polyhedra. For example, by Miller's rules, the triakis tetrahedron produces a total of 188 stellations, 136 of which are chiral. Using this same set of (Miller) rules, its dual polyhedron, the truncated tetrahedron, produces only 9 stellations, without including the truncated tetrahedron.

References

1. Sloane, N. J. A. (ed.). "SequenceA007304(Sphenic numbers: products of 3 distinct primes.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24.
2. Sloane, N. J. A. (ed.). "SequenceA003601(Numbers n such that the average of the divisors of n is an integer)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24.
3. Sloane, N. J. A. (ed.). "SequenceA000217(Triangular numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24.
4. "138". Numbers Aplenty. Retrieved 2023-07-24.
5. Sloane, N. J. A. (ed.). "SequenceA002858(Ulam numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24.
6. Sloane, N. J. A. (ed.). "SequenceA005101(Abundant numbers (sum of divisors of m exceeds 2m).)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24.
7. Sloane, N. J. A. (ed.). "SequenceA006991(Primitive congruent numbers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24.
8. Sloane, N. J. A. (ed.). "SequenceA095311(47-gonal numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-27.
9. Sloane, N. J. A. (ed.). "SequenceA014574(Average of twin prime pairs.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24.
10. Sloane, N. J. A. (ed.). "SequenceA006093(a(n) equal to prime(n) - 1.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24.
11. Sloane, N. J. A. (ed.). "SequenceA001097(Twin primes.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24.
12. Sloane, N. J. A. (ed.). "SequenceA034963(Sums of four consecutive primes.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
13. Sloane, N. J. A. (ed.). "SequenceA000010(Euler totient function phi(n): count numbers less than or equal to n and prime to n.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24.
14. Sloane, N. J. A. (ed.). "SequenceA002322(Reduced totient function psi(n))". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24.
15. Sloane, N. J. A. (ed.). "SequenceA027642(Denominator of Bernoulli number B_n.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24.
16. Sloane, N. J. A. (ed.). "SequenceA027641(Numerator of Bernoulli number B_n.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24.
17. Wenninger, Magnus J. (1983). "Chapter 3: Stellated forms of convex duals". Dual Models. Cambridge, UK: Cambridge University Press. pp. 36–37. doi:10.1017/CBO9780511569371. ISBN   9780521245241. MR   0730208. OCLC   8785984.
18. "Who's Afraid Of 138?!". Armada Music . Retrieved 2023-07-25.
19. "Who's Afraid Of 138?!". Beatport . Retrieved 2023-07-25.