138 (number)

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← 137 138 139 →
Cardinalone hundred thirty-eight
Ordinal138th
(one hundred thirty-eighth)
Factorization2 × 3 × 23
Divisors1, 2, 3, 6, 23, 46, 69, 138
Greek numeralΡろーΛらむだΗいーた´
Roman numeralCXXXVIII
Binary100010102
Ternary120103
Senary3506
Octal2128
DuodecimalB612
Hexadecimal8A16

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

In mathematics[edit]

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: .[a] 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 (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).[b]

The simplest Catalan solid, the triakis tetrahedron, produces 138 stellations (depending on rules chosen),[c] 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[edit]

In media[edit]

See also[edit]

Notes[edit]

  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[edit]

  1. ^ Sloane, N. J. A. (ed.). "Sequence A007304 (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.). "Sequence A003601 (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.). "Sequence A000217 (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.). "Sequence A002858 (Ulam numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-24.
  6. ^ Sloane, N. J. A. (ed.). "Sequence A005101 (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.). "Sequence A006991 (Primitive congruent numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-24.
  8. ^ Sloane, N. J. A. (ed.). "Sequence A095311 (47-gonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-27.
  9. ^ Sloane, N. J. A. (ed.). "Sequence A014574 (Average of twin prime pairs.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-24.
  10. ^ Sloane, N. J. A. (ed.). "Sequence A006093 (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.). "Sequence A001097 (Twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-24.
  12. ^ Sloane, N. J. A. (ed.). "Sequence A034963 (Sums of four consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  13. ^ Sloane, N. J. A. (ed.). "Sequence A000010 (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.). "Sequence A002322 (Reduced totient function psi(n))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-24.
  15. ^ Sloane, N. J. A. (ed.). "Sequence A027642 (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.). "Sequence A027641 (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.
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