70 is the fourth discretesphenic number, as the first of the form.[1] It is the smallestweird number, a natural number that isabundant but notsemiperfect,[2] where it is also the second-smallestprimitive abundant number, after20. 70 is in equivalence with the sum between the smallest number that is the sum oftwo abundant numbers, and the largest that is not (24,46).
70 is the tenthErdős–Woods number, since it is possible to find sequences of seventy consecutive integers such that each inner member shares afactor with either the first or the last member.[3][a] It is also the sixthPell number, preceding the tenth prime number29, in the sequence.
70 is the thirteenthhappy number indecimal, where7 is the first such number greater than 1 in base ten: the sum ofsquares of its digits eventually reduces to1.[7]
70 =2 ×5 × 7 simplifies to 7 ×10, or the product of the first happy prime in decimal, and the base (10).
Thecomposite index of 70 is 50,[8] which is the first non-trivial member of the 43-aliquot tree.
40, theEuler totient of100, is the second non-trivial member of the 43-aliquot tree.
The composite index of 100 is 74 (the aliquot part of 70),[8] the third non-trivial member of the 43-aliquot tree.
The sum 43 + 50 + 40 =133 represents the one-hundredth composite number,[8] where the sum of all members in this aliquot sequence up to 70 is the fifty-ninth prime,277 (this prime index value represents the seventeenth prime number and seventh super-prime,59).[9][5][c]
The sum of the first seven prime numbers aside from7 (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 eighthindexed primes (in thesequence of prime numbers) 13 + 17 + 19 is the seventhsquare number,49.
70 is the fourthcentral 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-bitbinary number for which half thebits are on, and half are off.[16]
In seven dimensions, the number oftetrahedral cells in a7-simplex is 70. This makes 70 the central element in a seven by sevenmatrix configuration of a 7-simplex in seven-dimensional space:
Aside from the 7-simplex, there are a total of seventy otheruniform 7-polytopes withsymmetry. The 7-simplex can be constructed as thejoin of apoint and a6-simplex, whoseorder is 7!, where the 6-simplex has a total of seventy three-dimensional and two-dimensionalelements (there are thirty-five3-simplex cells, and thirty-fivefaces that aretriangular).
70 is also the fifthpentatope number, as the number of 3-dimensional unit spheres which can be packed into a4-simplex (or four-dimensional analogue of theregular tetrahedron) of edge-length 5.[17]
InJewish tradition,Ptolemy II Philadelphus ordered 72 Jewish elders to translate theTorah intoGreek; the result was theSeptuagint (from theLatin 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:
UnderSocial 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).
Look upseventy in Wiktionary, the free dictionary.
Several languages, especially ones withvigesimal number systems, do not have a specific word for 70: for example,French:soixante-dix,lit. 'sixty-ten';Danish:halvfjerds, short forhalvfjerdsindstyve, 'three and a half score'. (For French, this is true only in France, Canada and Luxembourg; other French-speaking regions such asBelgium,Switzerland,Aosta Valley andJersey useseptante.)[18]
^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 = 24 × 3 × 7 × 11 × 13 × 19 × 23 × 29 × 37 × 43 × 47 × 53 × 67 × 73 × 2221, or approximately 2.62 × 1022.[4] Its largest prime factor is the sixty-seventhsuper-prime,[5] where 70 lies midway between the thirteenth pair ofsexy primes (67,73).[6]
^It is also a Harshad number in bases 6, 8, 9, 10, 11, 13, 14, 15 and 16.
^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".
