In the English language, the numbers 90 and 19 are often confused, as they sound very similar. When carefully enunciated, they differ in which syllable is stressed: 19 /naɪnˈtiːn/ vs 90 /ˈnaɪnti/. However, in dates such as1999, and when contrasting numbers in the teens and when counting, such as 17, 18, 19, the stress shifts to the first syllable: 19 /ˈnaɪntiːn/.
90 is apronic number as it is theproduct of9 and10,[1] and along with12 and56, one of only a few pronic numbers whose digits indecimal are also successive. 90 is divisible by the sum of itsbase-ten digits, which makes it the thirty-secondHarshad number.[2]
The number ofdivisors of 90 is 12.[7] Other smaller numbers with this property are60,72 and84. These four and96 are the five double-digit numbers with exactly 12 divisors.[8]
90 is the tenth and largest number to hold anEuler totient value of24;[9] no number has a totient that is 90, which makes it the eleventhnontotient (with50 the fifth).[10]
The twelfthtriangular number78[11] is the only number to have analiquot sum equal to 90, aside from thesquare of the twenty-fourth prime,892 (which iscentered octagonal).[12][13] 90 is equal to the fifth sum ofnon-triangular numbers, respectively between the fifth and sixth triangular numbers,15 and21 (equivalently16 + 17 ... + 20).[14] It is also twice45, which is the ninth triangular number, and the second-smallest sum of twelve non-zero integers, from two through thirteen.
90 can be expressed as the sum of distinct non-zerosquares in six ways, more than any smaller number (see image):[15]
.
90 as the sum of distinct nonzero squares
The square of eleven 112 = 121 is the ninetieth indexedcomposite number,[16] where the sum of integers is65, which in-turn represents the composite index of 90.[16] In thefractional part of thedecimal expansion of the reciprocal of11 inbase-10, "90" repeats periodically (when leading zeroes are moved to the end).[17]
The members of the firstprime sextuplet (7,11,13, 17,19,23) generate asum equal to 90, and the difference between respective members of the first and second prime sextuplets is also 90, where the second prime sextuplet is (97,101,103,107,109,113).[20][21] The last member of the second prime sextuplet, 113, is the 30thprime number. Since prime sextuplets are formed from prime members of lower orderprimek-tuples, 90 is also a record maximal gap between various smaller pairs of primek-tuples (which includequintuplets,quadruplets, andtriplets).[a]
This Witting configuration when reflected under thefinite space splits into 85 = 45 + 40 pointsand planes, alongside 27 + 90 + 240 = 357 lines.[25]
Whereas the rhombic enneacontahedron is thezonohedrification of the regular dodecahedron,[26] ahoneycomb of Witting polytopes holds verticesisomorphic to the E8lattice, whose symmetries can be traced back to the regular icosahedron via theicosian ring.[27]
The maximal number of pieces that can be obtained by cutting anannulus with twelve cuts is 90 (and equivalently, the number of 12-dimensionalpolyominoes that areprime).[28]
^90 is the record gap between the first pair ofprime quintuplets of the form(p,p+2,p+6,p+8,p+12) (A201073), while 90 is a record between the second and third prime quintuplets that have the form(p,p+4,p+6,p+10,p+12) (A201062). Regardingprime quadruplets, 90 is the gap record between the second and third set of quadruplets (A113404).Prime triplets of the form(p,p+4,p+6) have a third record maximal gap of 90 between the second and ninth triplets (A201596), and while there is no record gap of 90 for prime triplets of the form(p,p+2,p+6), the first and third record gaps are of6 and 60 (A201598), which are alsounitary perfect numbers like 90 (A002827).
