GOTW- Counting - Page 21

• 157 is a prime number. The next prime is 163 and the previous prime is 151, which with 157 forma sexy prime triplet. Taking the arithmetic mean of those primes yields 157, thus it is abalanced prime.
• In base 10, 157² is 24649. 158² is 24964, which uses the same digits as the previous square. 157 is the largest known number with a square that relates in this way to the square of the next integer
• 157 is a repdigit in base 12 (111)
• The simplest right angle triangle with rational sides that has area 157 has the longest side with a denominator of 45 digits.

• 158 is a nontotient, since there is no integer with 158 coprimes below it.
• 158 is a Perrin number, appearing after 68, 90, 119.
• 158 is the number of digits in the decimal expansion of 100!, or the product of all the natural numbers up to and including 100.

• 159 is the sum of 3 consecutive prime numbers: 47 + 53 + 59
• The previous prime number is 157, and the next is 163
• Given 159, the Mertens function returns 0. 159 is a Woodall number
• Clix spells the Roman numeral for 159 (CLIX)

• 160 is the sum of the first 11 primes, as well as the sum of the cubes of the first three primes
• Given 160, the Mertens function returns 0. 160 is the smallest number n with exactly 12 solutions to the equation f(x) = n
• The factors/divisors of 160 are: 1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 80, and 160

I'm moving on the the year

Year 161 (CLXI) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Caesar and Aurelius (or, less frequently, year 914 Ab urbe condita). The denomination 161 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years.

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Year 162 (CLXII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Rusticus and Plautius (or, less frequently, year 915 Ab urbe condita). The denomination 162 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years.

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By place

[edit]Roman Empire

• Lucius Verus begins a war with the Parthians, due to the invasion of Syria and Armenia by Vologases IV of Parthia.

[edit]By topic

[edit]Arts and sciences

• Arrian publishes Indica, a work on India and its people.

Year 163 (CLXIII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as theYear of the Consulship of Laelianus and Pastor (or, less frequently, year 916 Ab urbe condita). The denomination 163 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years.

[edit]Events

[edit]By place

[edit]Europe

• Statius Priscus reconquers Armenia; Artaxata is ruined.

Year 164 (CLXIV) was a leap year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as theYear of the Consulship of Macrinus and Celsus (or, less frequently, year 917 Ab urbe condita). The denomination 164 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years.

[edit]Events

[edit]By place

[edit]Roman Empire

• The governor of Syria, Avidius Cassius, one of Lucius Verus' generals, crosses the Euphrates and invades Parthian territory.
• Ctesiphon is captured by the Romans, but returned to the Parthians after the end of the war.
• The Antonine Wall in Scotland is abandoned by the Romans.
• A law is passed protecting property brought into a marriage by the wife.
• Marcus Aurelius gives his daughter Lucilla in marriage to his co-emperor Lucius Verus.
• Seleucia on the Tigris is destroyed.

Year 165 (CLXV) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Orfitus and Pudens (or, less frequently, year 918 Ab urbe condita). The denomination 165 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years.

[edit]Events

[edit]By place

[edit]Roman Empire

• A Roman military operation under Avidius Cassius is successful against Parthia, capturing Artaxata, Seleucia on the Tigris, and Ctesiphon. The Parthians sue for peace.
• A pandemic known as the Antonine Plague breaks out in Rome after the Roman army returns from Parthia. The plague significantly depopulates the Roman Empire.
• Legio II Italica is levied by emperor Marcus Aurelius.
• Dura-Europos is taken by the Romans.
• The Romans establish a garrison at Doura Europos on the Euphrates, a control point for the commercial route to the Persian Gulf.
• Avidius Cassius takes Nisibis and conquers the north of Mesopotamia.
• Marcus Aurelius creates four legal districts (iuridici) in Italy (5 if Rome is included).