View source for 70 (number)

{{Infobox number
| number = 70
| factorization=2 x 5 x 7
| divisor = 1, 2, 5, 7, 10, 14, 35, 70
| lang1 =  [[Hebrew numerals|Hebrew]]
| lang1 symbol = <span style="font-size:150%;">ע</span> 
| lang2 =  [[Lao numerals|Lao]]
| lang2 symbol = <span style="font-size:127%;">໗</span>
| lang3=[[Armenian numerals|Armenian]]|lang3 symbol=Հ
| lang4=[[Babylonian cuneiform numerals|Babylonian numeral]]|lang4 symbol=𒐕𒌋
| lang5=[[Egyptian numerals|Egyptian hieroglyph]]|lang5 symbol=<span style="font-size:300%;">𓎌</span>
}}

'''70''' ('''seventy''') is the natural number following [[69 (number)|69]] and preceding [[71 (number)|71]].

== Mathematics ==

=== Properties of the integer ===
'''70''' is the fourth discrete [[sphenic number]], as the first of the form <math>2 \times 5 \times r</math>.<ref>{{cite web|title=Sloane's A007304 : Sphenic numbers|url=https://oeis.org/A007304|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2016-05-29}}</ref> It is the smallest [[weird number]], a natural number that is [[Abundant number|abundant]] but not [[semiperfect]],<ref>{{cite web|title=Sloane's A006037 : Weird numbers|url=https://oeis.org/A006037|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2016-05-29}}</ref> where it is also the second-smallest [[primitive abundant number]], after [[20 (number)|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 (number)|24]], [[46 (number)|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 (arithmetic)|factor]] with either the first or the last member.<ref>{{cite web|title=Sloane's A059756 : Erdős-Woods numbers|url=https://oeis.org/A059756|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2016-05-29}}</ref>{{efn|1=The smallest sequence of seventy consecutive integers sharing a factor with either first or last member starts at the twenty-three digit number (with decimal representation), 26214699169906862478864 = 2<sup>4</sup> × 3 × 7 × 11 × 13 × 19 × 23 × 29 × 37 × 43 × 47 × 53 × 67 × 73 × 2221, or approximately 2.62 × 10<sup>22</sup>.<ref>{{Cite OEIS |A059757 |Initial terms of smallest Erdős-Woods intervals corresponding to the terms of A059756. |access-date=2024-07-31 }}</ref> Its largest prime factor is the sixty-seventh [[super-prime]],<ref name=A006450>{{Cite OEIS |A006450 |Prime-indexed primes: primes with prime subscripts. |access-date=2024-07-31 }}</ref> where 70 lies midway between the thirteenth pair of [[sexy prime]]s ([[67 (number)|67]], [[73 (number)|73]]).<ref>{{Cite OEIS |A023201 |Primes p such that p + 6 is also prime. (Lesser of a pair of sexy primes.) |access-date=2024-07-31 }}</ref> }} It is also the sixth [[Pell number]], preceding the tenth prime number [[29 (number)|29]], in the sequence <math>\{0, 1, 2, 5, 12, 29, \ldots\}</math>. 

70 is a [[palindromic number]] in bases 9 (77<sub>9</sub>), 13 (55<sub>13</sub>) and 34 (22<sub>34</sub>).{{efn|1=It is also a Harshad number in bases 6, 8, 9, 10, 11, 13, 14, 15 and 16. }}

==== Happy number ====

70 is the thirteenth [[happy number]] in [[decimal]], where [[7]] is the first such number greater than 1 in base ten: the sum of [[Square number|squares]] of its digits eventually reduces to [[1]].<ref>{{Cite OEIS |A007770 |Happy numbers: numbers whose trajectory under iteration of sum of squares of digits map (see A003132) includes 1. |access-date=2024-07-31 }}</ref>

70 = [[2]] × [[5]] × 7 simplifies to 7 × [[10]], or the product of the first happy prime in decimal, and the base (10).

==== Aliquot sequence ====
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,<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]], is the second 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 fifty-ninth prime, [[277 (number)|277]] (this prime index value represents the seventeenth prime number and seventh super-prime, [[59 (number)|59]]).<ref>{{Cite OEIS |A000040 |The prime numbers. |access-date=2024-07-31 }}</ref><ref name="A006450" />{{efn|1=Meanwhile, the [[aliquot sum]] of [[164 (number)|164]] = 74 + 40 + 50 is [[130 (number)|130]],<ref>{{Cite OEIS |A001065 |Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n. |access-date=2024-07-31 }}</ref> with a [[Divisor function|sum-of-divisor]]s of [[294 (number)|294]],<ref>{{Cite OEIS |A000203 |...the sum of the divisors of n. |access-date=2024-07-31 }}</ref> and an [[Arithmetic number|arithmetic mean of divisors]] of '''49'''.<ref>{{Cite OEIS |A003601 |Numbers n such that the average of the divisors of n is an integer: sigma_0(n) divides sigma_1(n). |access-date=2024-07-31 }}</ref><ref>{{Cite OEIS |A102187 |Arithmetic means of divisors of arithmetic numbers (arithmetic numbers, A003601, are those for which the average of the divisors is an integer). |access-date=2024-07-31 }}</ref> }}

==== Figurate numbers ====
* 70 is the seventh [[pentagonal number]].<ref>{{cite web|title=Sloane's A000326 : Pentagonal numbers|url=https://oeis.org/A000326|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2016-05-29}}</ref> 
* 70 is also the fourth 13-gonal ([[Polygonal number|tridecagonal]]) number.<ref>{{cite web|title=Sloane's A051865 : 13-gonal (or tridecagonal) numbers|url=https://oeis.org/A051865|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2016-05-29}}</ref>
*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 [[Sequence|indexed]] primes (in the [[List of prime numbers|sequence of prime numbers]]) 13 + 17 + 19 is the seventh [[square number]], [[49 (number)|49]].

==== Central binomial coefficient ====
70 is the fourth [[central binomial coefficient]], preceding <math>\{1, 2, 6, 20\}</math>, 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 [[bit]]s are on, and half are off.<ref>{{Cite OEIS |A000984 |Central binomial coefficients: binomial(2*n,n) as  (2*n)!/(n!)^2. }}</ref>

=== Geometric properties ===
==== 7-simplex ====
[[File:7-simplex t0.svg|left|thumb|Two-dimensional orthographic projection of the [[7-simplex]], a [[uniform 7-polytope]] with seventy [[Regular tetrahedron|tetrahedral cells]] ]]

In seven dimensions, the number of [[Tetrahedron|tetrahedral]] cells in a [[7-simplex]] is 70. This makes 70 the central element in a seven by seven [[Configuration (polytope)|matrix configuration]] of a 7-simplex in seven-dimensional space:

<math>\begin{bmatrix}\begin{matrix}8 & 7 & 21 & 35 & 35 & 21 & 7 \\ 2 & 28 & 6 & 15 & 20 & 15 & 6 \\ 3 & 3 & 56 & 5 & 10 & 10 & 5 \\ 4 & 6 & 4 & 70 & 4 & 6 & 4 \\ 5 & 10 & 10 & 5 & 56 & 3 & 3 \\ 6 & 15 & 20 & 15 & 6 & 28 & 2 \\ 7 & 21 & 35 & 35 & 21 & 7 & 8 \end{matrix}\end{bmatrix}</math>

Aside from the 7-simplex, there are a total of seventy other [[uniform 7-polytope]]s with <math>\mathrm {A_7}</math> [[A7 polytope|symmetry]]. The 7-simplex can be constructed as the [[Join (topology)|join]] of a [[Point (geometry)|point]] and a [[6-simplex]], whose [[Group order|order]] is 7!, where the 6-simplex has a total of seventy three-dimensional and two-dimensional [[Simplex#Elements|elements]] (there are thirty-five [[3-simplex]] cells, and thirty-five [[Face (geometry)|faces]] that are [[Equilateral triangle|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.<ref>{{cite web|title=Sloane's A000332 : Binomial coefficient binomial(n,4) = n*(n-1)*(n-2)*(n-3)/24|url=https://oeis.org/A000332|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2016-05-29}}</ref>

==== Leech lattice ====
The sum of the first 24 squares starting from 1 is 70{{sup|2}} = 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 religion ==
* In [[Judaism|Jewish]] tradition, [[Ptolemy II Philadelphus]] ordered 72 Jewish elders to translate the [[Torah]] into [[Greek language|Greek]]; the result was the [[Septuagint]] (from the [[Latin language|Latin]] for "seventy"). The Roman numeral seventy, LXX, is the scholarly symbol for the Septuagint.

* 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 other fields ==
* In some traditions, 70 years of marriage is marked by a [[platinum]] [[wedding anniversary]].
* 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 ==
{{Main article|Numeral (linguistics)}}
{{Wiktionary|seventy}}

Several languages, especially ones with [[vigesimal]] number systems, do not have a specific word for 70: for example, {{Langx|fr|soixante-dix|lit=sixty-ten}}; {{Langx|da|halvfjerds}}, short for {{Langx|da|halvfjerdsindstyve|lit=three and a half score|label=none}}. (For French, this is true only in France, Canada and Luxembourg; other French-speaking regions such as [[Belgium]], [[Switzerland]], [[Aosta Valley]] and [[Jersey]] use {{Lang|fr|septante}}.)<ref>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".</ref>

== Notes ==
{{Notelist}}

== References ==
<!--This article uses the Cite.php citation mechanism. If you would like more information on how to add references to this article, please see http://meta.wikimedia.org/wiki/Cite/Cite.php -->
{{Reflist}}

== External links ==
{{Integers|zero}}
{{Authority control}}

{{DEFAULTSORT:70 (Number)}}
[[Category:Integers]]

[[la:Septuaginta]]