138 Jean-Marie De Koninck

1 880

• the smallest happy number n such that n+1 and n+2 are also happy: a number

is said to be happy if the iteration process of summing the squares of its digits

leads eventually to 1 (otherwise the process leads to 4): the sequence of happy

numbers begins as follows: 1, 7, 10, 13, 19, 23, 28, 31, 32, 44, . . . (R.K. Guy

[101],

E34)136.

1 885

• the third composite number n such that σ(n + 8) = σ(n) + 8 (see the number

1 615);

• the fifth (and largest known) number n such that 2n−1 + n is a prime number

(see the number 237).

1 886 (= 2 · 23 · 41)

• the smallest square-free composite number n such that p|n =⇒ p + 4|n + 4; the

sequence of numbers satisfying this property begins as follows: 1 886, 50 711,

149 171, 222 101, 628 421, 766 931, . . . (see the number 399).

1 888

• the fourth solution of σ(n) = 2n + 4; the sequence of numbers satisfying this

equation begins as follows: 12, 70, 88, 1 888, 4 030, 5 830, 32 128, 521 728,

1 848 964, 8 378 368, . . .

137

1 889

• the smallest prime number p such that Ω(p − 1) = Ω(p + 1) = 6: here 1 888 =

25 · 59 and 1 890 = 2 · 33 · 5 · 7 (see the number 271).

1 905

• the fourth Euler pseudoprime in base 2, that is an odd number n such that

2(n−1)/2

≡ (

2

n

) (mod n) (where (

a

n

) stands for the Jacobi Symbol); the se-

quence of numbers satisfying this property begins as follows: 561, 1105, 1729,

1905, 2047, 2465, 4033, 4681, 5461, 6601, 8321, 8481, 10261, 10585, 12801,

15709, 15841, 16705, 18705, . . .

136A simple observation: it seems that roughly one seventh of the numbers are happy !

137It

is easy to show that each number n =

2α

· p, where α is a positive integer such that

p = 2α+1 − 5 is prime, is a solution of σ(n) = 2n + 4; this is the case in particular when

α = 2, 3, 5, 7, 9, 11, 17, 19, 25, 31, 35, 55, 65 (and for no other values of α ≤ 100); the solutions corre-

sponding to α = 2, 3, 5, 7, 9, 11 are included in the above list.