# 38

This number is a composite. The magic constant in the only possible magic hexagon (which utilizes all the natural integers up to and including the prime number 19). The sum of squares of the first three primes (22 + 32 + 52). [Kimberling] The number of Mersenne primes known to humanity prior to the 21st century. The sum of the first 38 cubed primes (23 + 33 + ... + 1633) is prime. [De Geest] 38 = 2 * 19. Note that 32 + 819 is prime. [Honaker] 38 * 238 + 38 + 1 is prime. [Noll] 38 of the first one hundred primes are full period primes. Note that it is conjectured that about 3/8 of all primes are full period primes. The largest known even number which can be represented as sum of two distinct primes in only one way (i.e., 38 = 31 + 7). [Gupta] Let spf = sum of prime factors (where spf(p) = 0); then spf(38) = 21, spf(21) = 10, spf(10) = 7, and spf(7) = 0. Note that 21 + 10 + 7 + 0 = 38. The numbers 20 and 74 also share this property. [Moseley] 38 = 3(π(3)) + 8(π(8)). [Firoozbakht] The largest even number such that every partition into two odd parts contains at least one prime. [Brahinsky] 38 is the average of first primes up to 83. [Silva] Smallest composite number n whose number of circular loops equals Omega(n), i.e., the number of prime divisors of n counted with multiplicity. Circular loops occur in the digits 0, 6, 8, or 9 only. Note that the digit 8 contains two loops. The sequence begins 38, 58, 68, 69, ... . [Honaker] 38 followed by any number of 1's is composite. [Luen]

(There are 2 curios for this number that have not yet been approved by an editor.)