70 (number): Difference between revisions
→Aliquot sequence: out of note, and sum of all members |
|||
Line 31: | Line 31: | ||
70 contains an [[aliquot sum]] of [[74 (number)|74]], in an [[aliquot sequence]] of four composite numbers (70, 74, [[40 (number)|40]], [[50 (number)|50]], [[43 (number)|43]]) in the prime [[43 (number)|43]]-aliquot tree. |
70 contains an [[aliquot sum]] of [[74 (number)|74]], in an [[aliquot sequence]] of four composite numbers (70, 74, [[40 (number)|40]], [[50 (number)|50]], [[43 (number)|43]]) in the prime [[43 (number)|43]]-aliquot tree. |
||
* The [[Composite number|composite index]] of 70 is 50, which is the first non-trivial member of the 43-aliquot tree. |
* The [[Composite number|composite index]] of 70 is 50,<ref name="A002808">{{Cite OEIS |A002808 |The composite numbers. |access-date=2024-07-31 }}</ref> which is the first non-trivial member of the 43-aliquot tree. |
||
* 40, the [[Euler totient]] of [[100]] (as twice 50), is the second non-trivial member of the 43-aliquot tree. |
* 40, the [[Euler totient]] of [[100]] (as twice 50), is the second non-trivial member of the 43-aliquot tree. |
||
* The composite index of 100 is 74 (the aliquot part of 70), the third non-trivial member of the 43-aliquot tree. |
* The composite index of 100 is 74 (the aliquot part of 70),<ref name="A002808" /> the third non-trivial member of the 43-aliquot tree. |
||
The sum 43 + 50 + 40 = [[133 (number)|133]] represents the one-hundredth composite number,<ref name="A002808" /> where the sum of all members in this aliquot sequence up to 70 is the forty-ninth prime, [[277 (number)|277]].<ref>{{Cite OEIS |A000040 |The prime numbers. |access-date=2024-07-31 }}</ref> |
|||
==== Central binomial coefficient ==== |
==== Central binomial coefficient ==== |
Revision as of 18:44, 31 July 2024
| ||||
---|---|---|---|---|
Cardinal | seventy | |||
Ordinal | 70th (seventieth) | |||
Factorization | 2 × 5 × 7 | |||
Divisors | 1, 2, 5, 7, 10, 14, 35, 70 | |||
Greek numeral | Ο´ | |||
Roman numeral | LXX | |||
Binary | 10001102 | |||
Ternary | 21213 | |||
Senary | 1546 | |||
Octal | 1068 | |||
Duodecimal | 5A12 | |||
Hexadecimal | 4616 | |||
Hebrew | ע | |||
Lao | ໗ | |||
Armenian | Հ | |||
Babylonian numeral | 𒐕𒌋 | |||
Egyptian hieroglyph | 𓎌 |
70 (seventy) is the natural number following 69 and preceding 71.
70 is the value whose factorial is closest to a googol, where .
Mathematics
Properties of the integer
70 is the fourth discrete sphenic number, as the first of the form .[1] It is the smallest weird number, a natural number that is abundant but not semiperfect,[2] where it is also the second-smallest primitive abundant number, after 20. 70 is in equivalence with the sum between the smallest number that is the sum of two abundant numbers, and the largest that is not (24, 46).
70 is the tenth Erdős–Woods number, since it is possible to find sequences of seventy consecutive integers such that each inner member shares a factor with either the first or the last member.[3] It is also the sixth Pell number, preceding the tenth prime number 29, in the sequence .
70 is a palindromic number in bases 9 (779), 13 (5513) and 34 (2234).[a]
Figurate numbers
- 70 is the seventh pentagonal number.[4]
- 70 is also the fourth 13-gonal (tridecagonal) number.[5]
- 70 is the fifth pentatope number.
The sum of the first seven prime numbers aside from 7 (i.e., 2, 3, 5, 11, …, 19) is 70; the first four primes in this sequence sum to 21 = 3 × 7, where the sum of the sixth, seventh and eighth indexed primes (in the sequence of prime numbers) 13 + 17 + 19 is the seventh square number, 49.[b]
Aliquot sequence
70 contains an aliquot sum of 74, in an aliquot sequence of four composite numbers (70, 74, 40, 50, 43) in the prime 43-aliquot tree.
- The composite index of 70 is 50,[6] which is the first non-trivial member of the 43-aliquot tree.
- 40, the Euler totient of 100 (as twice 50), is the second non-trivial member of the 43-aliquot tree.
- The composite index of 100 is 74 (the aliquot part of 70),[6] the third non-trivial member of the 43-aliquot tree.
The sum 43 + 50 + 40 = 133 represents the one-hundredth composite number,[6] where the sum of all members in this aliquot sequence up to 70 is the forty-ninth prime, 277.[7]
Central binomial coefficient
70 is the fourth central binomial coefficient, preceding , as the number of ways to choose 4 objects out of 8 if order does not matter; this is in equivalence with the number of possible values of an 8-bit binary number for which half the bits are on, and half are off.[8]
Geometric properties
7-simplex
In seven dimensions, the number of tetrahedral cells in a 7-simplex is 70. This makes 70 the central element in a seven by seven matrix configuration of a 7-simplex in seven-dimensional space:
Aside from the 7-simplex, there are a total of seventy other uniform 7-polytopes with symmetry. Worth mentioning, the 7-simplex can be constructed as the join of a point and a 6-simplex, whose order is 7!. The 6-simplex, in particular, has a total of seventy three-dimensional and two-dimensional elements (where there are thirty-five 3-simplex cells, and thirty-five faces that are triangular).
70 is also the fifth pentatope number, as the number of 3-dimensional unit spheres which can be packed into a 4-simplex (or four-dimensional analogue of the regular tetrahedron) of edge-length 5.[9]
Leech lattice
The sum of the first 24 squares starting from 1 is 702 = 4900, i.e. a square pyramidal number. This is the only non trivial solution to the cannonball problem, and relates 70 to the Leech lattice in twenty-four dimensions and thus string theory.
In science
- 70 is the atomic number of ytterbium, a lanthanide
Astronomy
- Messier object M70, a magnitude 9.0 globular cluster in the constellation Sagittarius
- The New General Catalogue object NGC 70, a magnitude 13.4 spiral galaxy in the constellation Andromeda
In religion
- In Jewish tradition:
- Seventy souls went down to Egypt to begin the Hebrews' Egyptian exile (Genesis 46:27).
- There is a core of 70 nations and 70 world languages, paralleling the 70 names in the Table of Nations.
- There were 70 men in the Great Sanhedrin, the Supreme Court of ancient Israel. (Sanhedrin 1:4.)
- According to the Jewish Aggada, there are 70 perspectives ("faces") to the Torah (Numbers Rabbah 13:15).
- Seventy elders were assembled by Moses on God's command in the desert (Numbers 11:16–30).
- Psalm 90:10 allots three score and ten (70 years) for a man's life, and the Mishnah attributes that age to "strength" (Avot 5:32), as one who survives that age is described by the verse as "the strong".
- Ptolemy II Philadelphus ordered 72 Jewish elders to translate the Torah into Greek; the result was the Septuagint (from the Latin for "seventy"). The Roman numeral seventy, LXX, is the scholarly symbol for the Septuagint.
- In Christianity:
- In Matthew 18:21–22, Jesus tells Peter to forgive people seventy times seven times.
- In Luke 10:1–24, Jesus appoints Seventy Disciples and sends them out in pairs to preach the Gospel.
- Seventy is a priesthood office in the Latter Day Saint religion.
- In Islamic history and in Islamic interpretation the number 70 or 72 is most often and generally hyperbole for an infinite amount:
- There are 70 dead among the Prophet Muhammad's adversaries during the Battle of Badr.
- 70 of the Prophet Muhammad's followers are martyred at the Battle of Uhud.
- In Shia Islam, there are 70 martyrs among Imam Hussein's followers during the Battle of Karbala.
In law
- In certain cases, copyrights expire after 70 years.
In sports
- In Olympic archery, the targets are 70 meters from the archers.
- In college football, two teams scored 70 points in bowl games, the most in such contests: first the West Virginia Mountaineers against the Clemson Tigers in the 2012 Orange Bowl, and the Army Black Knights against the Houston Cougars in the 2018 Armed Forces Bowl.
- The number of the laps of the Canadian Grand Prix and Hungarian Grand Prix.
In other fields
- 70 miles per hour is a common speed limit for freeways in many American states, primarily in the central United States (in the Eastern U.S. the speed limit is generally 65, in the Western U.S. it is 75).
- 70 miles per hour is the national speed limit in the United Kingdom for cars and motorcycles on the best grades of road.[10]
- 70 years of marriage is marked by a platinum wedding anniversary.
- 70 is the hull number of the U.S. Navy's nuclear aircraft carrier USS Carl Vinson (CVN-70), named after U.S. Representative Carl Vinson.
- The French department Haute-Saône is number 70.
- As a year, "70" may refer to 70 BC, AD 70, or 1970.
- The number 70 is frequently referenced by the musical duo Boards of Canada: they have songs titled "Sixtyten" (Music Has the Right to Children, 1998) and "The Smallest Weird Number" (Geogaddi, 2002), and their record label is named Music70.
- Under Social Security (United States), the age at which a person can receive the maximum retirement benefits (and may do so and continue working without reduction of benefits)
Number name
Several languages, especially ones with vigesimal number systems, do not have a specific word for 70: for example, French: soixante-dix, lit. 'sixty-ten'; Danish: halvfjerds, short for halvfjerdsindstyve, 'three and a half score'. (For French, this is true only in France; other French-speaking regions such as Belgium, Switzerland, Aosta Valley and Jersey use septante.[11])
Notes
- ^ It is also a Harshad number in bases 6, 8, 9, 10, 11, 13, 14, 15 and 16.
- ^ 70 is a happy number in decimal.
References
- ^ "Sloane's A007304 : Sphenic numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
- ^ "Sloane's A006037 : Weird numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
- ^ "Sloane's A059756 : Erdős-Woods numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
- ^ "Sloane's A000326 : Pentagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
- ^ "Sloane's A051865 : 13-gonal (or tridecagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
- ^ a b c Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-07-31.
- ^ Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-07-31.
- ^ Sloane, N. J. A. (ed.). "Sequence A000984 (Central binomial coefficients: binomial(2*n,n) as (2*n)!/(n!)^2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ "Sloane's A000332 : Binomial coefficient binomial(n,4) = n*(n-1)*(n-2)*(n-3)/24". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
- ^ The Official Highway Code, pub. Department for Transport (Revised 2007 Edition). ISBN 978-0-11-552814-9. A white circular sign with a black diagonal stripe indicates that the national speed limit applies. This depends on the vehicle type and grade of road. The table on p. 40 shows the highest speed permitted to be 70 mph, for normally-laden cars and motorcycles on dual-carriageways and motorways.
- ^ Peter Higgins, Number Story. London: Copernicus Books (2008): 19. "Belgian French speakers however grew tired of this and introduced the new names septante, octante, nonante etc. for these numbers".
External links
- On some Philological Peculiarities in the English Authorized Version of the Bible. By Thomas Watts, Esq.