>>1404029169 is the sixth integer n, bigger than all the other integer n, such that the fractional part of (3/2)^n is closer to 1/2 than (3/2)^k for all integer values of k between 1 and n. 420 is the seventh.
ln(69!)/69-e/10 is a very good approximation of 3.
69 is the smallest positive integer x such that there exists integers y and z such that 2+x^4=y^2+z^3
Everyone knows 163 is based, but I know the sum of the first 163 prime numbers is divisible by 69. The sum of the first 69 prime numbers is 10538 which is divisible by 479 (a prime). The first five digits of 479/69 are 6.9420. The other two prime factors of 10538 are 11 and 2. 11/69 has 4, 2 and 0 as the 5-7th digits.
floor(69.69*ln(420))=420
The sum of the divisors of 69 is 96. In other words, 69 is the first number greater than one such that sigma(n)=reversal(n). 69 is also the first number greater than one such that sigma(n) contains the same digits as n in base 10. Also 96 is the smallest integer n such that (6n*666-3)*2^666-1 is a prime number. 697 is the smallest integer such that (6n*6977-3)*2^6977-1 is a prime number.
Fun fact: 6977 is prime and 697 is semiprime
69 can be written as the average of 2 adjacent prime numbers. In fact 420 can be written as the average of 2 adjacent prime numbers too.
Phi(prime(69))=prime(phi(69)). The phi function counts the amount of integers that are coprime with n.
69 is the smallest positive number n such that applying n->prime(n) 6 times results in a number which ends with n (prime(prime(prime(prime(prime(prime(69)))))) = 54615469 which ends in 69).
69 is the first number k greater than 1 where the fractional part of (101/100)^k is greater than 1-1/k. The next is 180.
