GOTW- Counting - Page 13

101 is the 26th prime number and a palindromic number (and so a palindromic prime). The next prime is 103, with which it makes a twin prime pair, making 101 a Chen prime. Because the period length of its reciprocal is unique among primes, 101 is a unique prime. 101 is anEisenstein prime with no imaginary part and real part of the form . 101 is the only prime with alternating 1 s and 0 s.

101 is the sum of five consecutive primes (13 + 17 + 19 + 23 + 29). Given 101, the Mertens function returns 0. 101 is the fifth alternating factorial.

101 is a centered decagonal number.

The decimal representation of 2^101-1 is 2 535301 200456 458802 993406 410751. The prime decomposition of that is 7 432339 208719 x 341117 531003 194129 ^[1]

For a 3-digit number in base 10, this number has a relatively simple divisibility test. The candidate number is split into groups of four, starting with the rightmost four, and added up to produce a 4-digit number. If this 4-digit number is of the form 1000a + 100b + 10a + b (where a and bare integers from 0 to 9), such as 3232 or 9797, or of the form 100b + b, such as 707 and 808, then the number is divisible by 101. This might not be as simple as the divisibility tests for numbers like 3 or 5, and it might not be terribly practical, but it is simpler than the divisibility tests for other 3-digit numbers.^{[citation needed]}

On the seven-segment display of a calculator, 101 is both a strobogrammatic prime and a dihedral prime.

Sheils i already got his number many times 😆 😆

102 is an abundant number and semiperfect number. It is a sphenic number. It is the sum of four consecutive prime numbers (19 + 23 + 29 + 31).

The sum of Euler's totient function f(x) over the first eighteen integers is 102.

102 is the third base 10 polydivisible number, since 1 is divisible by 1, 10 is divisible by 2 and 102 is divisible by 3. This also shows that 102 is a Harshad number. 😆

One hundred [and] four is a primitive semiperfect numberand a composite number, with its divisors being 1, 2, 4, 8, 13, 26, 52 and 104. As it has 8 divisors total, and 8 is one of those divisors, 104 is a refactorable number. The distinct prime factors of 104 add up to 15, and so do the ones of 105, hence the two numbers form a Ruth-Aaron pair under the first definition.

In regular geometry, 104 is the smallest number of unit line segments that can exist in a plane with four of them touching at every vertex.

105 is a triangular number, a 12-gonal number and a Zeisel number. It is a sphenic number, and is the product of three consecutive prime numbers. 105 is the double factorial of 7. It is also the sum of the first five square pyramidal numbers.

105 comes in the middle of the prime quadruplet (101, 103, 107, 109). The only other such odd numbers less than a thousand are 15, 195 and 825. 105 is also a pseudoprime to the prime bases 13, 29, 41, 43, 71, 83 and 97. The distinct prime factors of 105 add up to 15, and so do those of 104, hence the two numbers form a Ruth-Aaron pair under the first definition.

105 is also a number n for which is prime, for . (This even works up to , ignoring the negative sign.)

105 is the smallest integer such that the factorization of over Q includes coefficients other than .

One hundred [and] seven is the 28th prime number. The next prime is 109, with which it comprises a twin prime, making 107 a Chen prime.

Plugged into the equation , 107 yields 162259276829213363391578010288127, a Mersenne prime. 107 is itself a safe prime.

One hundred [and] eight (or nine dozen) is an abundant number and a semiperfect number. It is a tetranacci number.

It is the hyperfactorial of 3 since it is of the form .

108 is a number that is divisible by the value of its f function, which is 36. 108 is also divisible by the total number of its divisors (12), hence it is a refactorable number.

In Euclidean space, the interior angles of a regular pentagon measure 108 degrees each.

There are 108 free polyominoes of order 7.

In base 10, it is a Harshad number and a self number.

One hundred [and] nine is the 29th prime number, and also a Chen prime. The previous prime is 107, making them both twin primes. 109 is a centered triangular number.

109 is the smallest factor of one more than the product of the first twenty-three terms of the Euclid–Mullin sequence, making it the twenty-fourth term.

109 is the smallest number which is palindromic in bases 5 and 9.

The cycle of the decimal equivalent of 1/109 ends ...853211, the first six Fibonacci numbers in reversed order.