The Penguin Dictionary of Curious and Interesting Numbers

28

Perfect numbers correspond one-for-one with the Mersenne primes.
Researchers, without having produced any odd perfects, have discovered a great deal about them, if it makes sense to say that you know a great deal about something that may not exist.

30

Primorials

Primorial p, denoted p#, is defined only if p is prime. It is then equal to the product of all the primes up to and including p. 30 = 5# = 5 x 3 x 2.
The sequence of primorials starts: 1 2 6 30 210 2310 30,030 510,510 9,699,690 223,092,870...

31

n# + 1 is prime for 2, 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, and for no other values below 3088.

46

A famous, or infamous, example of numerology: in Psalm 46, the 46th word is 'shake'. The 46th word from the end counting backwards is 'spear'. Shakespear! Why? Well, when the King James Authorized Version was completed in 1610 (= 35 x 46), Shakespear was 46 years old!

51

This appears to be the first uninteresting number, which of course makes it an especially interesting number, because it is the smallest number to have the property of being uninteresting.
It is therefore also the first number to be simultaneously interesting and uninteresting.

127

127 = 2⁷ - 1 is the 7th Mersenne number, denoted M₇, and the 4th Mersenne prime, and therefore the source of the 4th perfect number.

1001

1001 = 7 x 11 x 13
This is the basis for a test of divisibility that will test for all three divisors simultaneously. Mark off the numbers to be tested in groups of 3 digits from the unit position. [...] for example 68,925,857. Add the 1st, 3rd, 5th groups and take away the total of the 2nd, 4th ... groups. The number will be divisible by 7, 11 or 13, if the result is divisible by 7, 11 or 13 respectively: in this example, 68 + 857 - 925 = 0.

27,594

This number can be written in 2 curiously related ways as a product: 27594 = 73 x 9 x 42 = 7 x 3942 . [Madachy]

111,777

This is 'the least integer not nameable in fewer than nineteen syllables', and yet it has just been defined (apparently) in eighteen syllables. This is Berry's paradox. [Russell and Whitehead, Principia Mathematica]

277,777,788,888,899

[...] This is the smallest number with persistence 11 [...] There is no number less than 10⁵⁰ with persistence greater than 11.