The Lost Numbers: 4 8 15 16 23 42

In the TV-series "Lost", Episode 18, the plot revolves around six numbers. This page deals with their mathematical significance. This page does not deal with occurences of the numbers within the show, within other works of fiction or within other nonmathematical works of nonfiction, nor does it cover occult interpretations of their meaning. For pages that address those issues Google is your friend.

Summary

The Lost Numbers are not part of an integer sequence that is known and useful in many engineering or mathematical applications, but it is a part of these lesser known integer sequences:

A122115
A130826

Note that both of these sequences appeared on OEIS after the episode aired, so I don't think the show's creators were aware of them. Being part of a well known TV show is however notable in its own right so now there is an integer sequence in OEIS dedicated to the Lost Numbers:

A104101

Are they prime numbers?

One of them, 23, is indeed a prime number. The others are not. For your convenience, here are the prime factorizations:

4	=	2²
8	=	2³
15	=	3*5
16	=	2⁴
23	=	23 (prime)
42	=	237
108	=	2²*3³
741880	=	2¹⁰3²5723

For reference:

A prime number is a natural number, greater than 1, that is only divisible with itself and one.
Every number greater than 1 is either a prime or can be expressed as a product of primes.

The prime numbers are sequence A000040 in OEIS.

Are they perfect numbers?

No. The first perfect numbers are 6, 28 and 496. But there are 6 numbers so out of coincidence the number of numbers is a perfect number.
For reference: A perfect number is a number that is the same as the sum of its divisors (except itself) for example 6=1+2+3 and 28=1+2+4+7+14.
The perfect numbers are sequence A000396 in OEIS.

Are they decimals of π?

No, at least not in the first 3 200 000 000 digits, according to pisearch.
The decimal expansion of π is sequence A000796 in OEIS.

Can we leave out a number?

4, 8, 15, 16 and 23 are part of the two previously known sequences:
A084345, the numbers with non-prime number of 1:s in their binary expansion.
A084561, the numbers with square number of 1:s in their binary expansion.
For your convenience, here are all binary expansions (and their hex representation):

    dec    binary    hex
      4 = %0000100 = 0x04
      8 = %0001000 = 0x08
     15 = %0001111 = 0x0F
     16 = %0010000 = 0x10
     23 = %0010111 = 0x17
     42 = %0101010 = 0x2A
    108 = %1101100 = 0x6C
7418880 = %11100010011010000000000 = 0x713400

Mathematics is a huge and fascinating subject. Good luck and have fun!

Links

Mathworld - The best general mathematics resource.
OEIS - The best integer sequence resource.
Insultingly stupid movie physics - High quality commentary on science in movies.
My home page.
https://oeis.org/A130826