Counting game

[Mone]

37
It is a prime number
Prime number
In mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. The first twenty-six prime numbers are:An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC. The number 1 is by definition not a prime number...
the fifth lucky prime, the first irregular prime, the third unique prime
Unique prime
In mathematics, a unique prime is a certain kind of prime number. A prime p ≠ 2, 5 is called unique if there is no other prime q such that the period length of the decimal expansion of its reciprocal, 1 / p, is equivalent to the period length of the reciprocal of q, 1 / q...
and the third cuban prime
Cuban prime
A cuban prime is a prime number that is a solution to one of two different specific equations involving third powers of x and y. The first of these equations is:and the first few cuban primes from this equation are :...
of the form

It is a factor of all 3-digit base 10 repdigit
Repdigit
In recreational mathematics, a repdigit is a natural number composed of repeated instances of the same digit, most often in the decimal numeral system....
such as 111. 37 is the smallest prime that is not also a supersingular prime
Supersingular prime
In mathematics, a supersingular prime is a prime number satisfying one of the following concepts:*Supersingular prime , in algebraic number theory*Supersingular prime...

It is a centered hexagonal number
Centered hexagonal number
A centered hexagonal number, or hex number, is a centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice....
and a star number
Star number
A star number is a centered figurate number that represents a centered hexagram, such as the one that Chinese checkers is played on.The nth star number is given by the formula 6n + 1..

37 and 38 are the first pair of consecutive positive integers not divisible by any of their digits.

Every positive integer is the sum of at most 37 fifth powers (see Waring's problem
Waring's problem
In number theory, Waring's problem, proposed in 1770 by Edward Waring, asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most s kth powers of natural numbers...

37 appears in the Padovan sequence
Padovan sequence
The Padovan sequence is the sequence of integers P defined by the initial valuesand the recurrence relationThe first few values of P are...
preceded by the terms 16, 21, and 28 (it is the sum of the first two of these).

Since the greatest prime factor of 372 + 1 = 1370 is 137, which is obviously more than 37 twice, 37 is a Størmer number
Størmer number
In mathematics, a Størmer number or arc-cotangent irreducible number, named after Carl Størmer, is a positive integer n for which the greatest prime factor of n2 + 1 meets or exceeds 2n. The first few Størmer numbers are 1, 2, 4, 5, 6, 9, 10, 11, 12, 14, 15, 16, 19, 20, etc...
37 is the only two digit number in base 10 whose product, when multiplied by two, subtracted by one, and then read backwards, equals the original two digit number: 37x2=74-1=73, 73 backwards is 37.