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 =

· 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.
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