GOTW- Counting - Page 11

Eighty-one is the square of 9 and the fourth power of 3. Like all powers of three, 81 is a perfect totient number. It is a heptagonal number and a centered octagonal number. It is also a tribonacci number, and an open meandric number. 81 is the ninth member of the Mian-Chowla sequence.

In base 10, it is a Harshad number, and one of three non-trivial numbers (the other two are 1458 and 1729) which, when its digits are added together, produces a sum which, when multiplied by its reversed self, yields the original number:

8 + 1 = 9

9 9 = 81

(although this case is somewhat degenerate, as the sum has only a single digit).

Look up eighty-one in Wiktionary, the free dictionary.

The inverse of 81 is 0.012345679 recurring, missing only the digit "8" from the complete set of digits. This is an example of the general rule that, in base b,

omitting only the digit b-2.

Eighty-two is the twenty-third biprime and the twelfth of the form (2.q). The aliquot sum of 82 is 44 within the aliquot sequence (82,44,40,50,43,1,0)) 82 being the eighth composite number in the 43-aliquot tree.

Eighty-two is a companion Pell number.

Eighty-two is a happy number

Eighty-three is the sum of three consecutive primes (23 + 29 + 31) as well as the sum of five consecutive primes (11 + 13 + 17 + 19 + 23).

It is also the 23rd prime number, following 79 and preceding 89. 83 is a Sophie Germain prime and a safe prime, and also a Chen prime. It is an Eisenstein prime with no imaginary part and real part of the form .

It is also a highly cototient number.

Eighty-four is the sum of the first seven triangular numbers (making it a tetrahedral number), as well as the sum of a twin prime (41 + 43). Being thrice a perfect number, 84 is itself a semiperfect number.

Eighty-four is also the lim sup of the largest finite subgroup of the mapping class group of a genus surface divided by .

A hepteract is a seven-dimensional hypercube with 84 penteract 5-faces

86 is the 25th distinct semiprime and the 13th of the form (2.q). The aliquot sum of 86 is 46 within the aliquot sequence(86,46,26,16,15,9,4,3,1,0) 86 being the 17th composite number in the 3-aliquot tree.

86 is the middle number in the second cluster of three discrete semiprimes between 85 and 87 being themselves discrete semiprimes.

86 is a semiprime, nontotient, a noncototient, a happy number, and a self number. It appears in the Padovan sequence, preceded by the terms 37, 49, 65 (it is the sum of the first two of these).

Since it is possible to find sequences of 86 consecutive integers such that each inner member member shares a factor with either the first or the last member, 86 is an Erdos–Woods number.

86 is a repdigit in base 6 (222).

Eighty-seven is the twenty-sixth distinct semiprime and the eighth of the form (3.q). The aliquot sum of 87 is 33 within the aliquot sequence (87,33,15,9,4,3,1,0) 86 being the eighteenth composite number in the 3-aliquot tree.

Eighty-seven is the sum of the squares of the first four primes. It is also the sum of the sum of divisors of the first ten integers.

Eighty-Eight is a refactorable number, a primitive semiperfect number and an untouchable number. It is also an hexadecagonal number. 😆

Eighty-nine is the 24th prime number, following 83 and preceding 97. 89 is a Chen prime. It is the smallest Sophie Germain prime to start a Cunningham chain of the first kind of six terms, {89, 179, 359, 719, 1439, 2879}. 89 is an Eisenstein prime with no imaginary part and real part of the form .

89 is a Fibonacci number. The first few digits of its reciprocal coincide with the Fibonacci sequence due to the identity

89 is also a Markov number, appearing in solutions to the Markov Diophantine equation with other odd-indexed Fibonacci numbers.

Although 89 is not a Lychrel number in base 10, it is unusual that it takes 24 iterations of the reverse and add process to reach a palindrome. Among the known non-Lychrel numbers in the first 10000 integers, no other number requires that many or more iterations. The palindrome reached is also unusually large.^[1]

• Because 90 is the sum of its unitary divisors (excluding itself), it is a unitary perfect number, and because it is equal to the sum of a subset of its divisors, it is a semiperfect number. 90 is a pronic number. But it is also a nontotient. It is a Perrin number, preceded in the sequence by 39, 51, 68.

• In normal space, the interior angles of a square measure 90 degrees each. Also, in a right triangle, the angle opposing the hypotenuse measures 90 degrees, with the other two angles adding up to 90 for a total of 180 degrees.^[1] Thus, an angle measuring 90 degrees is called a right angle.^[2]

• 90 is divisible by the sum of its base 10 digits, thus it is a Harshad number.