134 Jean-Marie De Koninck

1 674

• the smallest number n which allows the sum

i≤n

1

i

to exceed 8 (see the number

83).

1 675

• the third number which does not produce a palindrome by the 196-algorithm

(see the number 196).

1 676

• the seventh number n 9 such that n =

∑r

i=1

di,

i

where d1, . . . , dr stand for

the digits of n: here 1 676 =

11

+

62

+

73

+

64

(see the number 175).

1 680 (=

24

· 3 · 5 · 7)

• the 17th highly composite number (see the number 180).

1 681 (= 412)

• the smallest composite number of the form

n2

+ n + 41 (with n = 40);

• the smallest composite number of the form

n2

− 79n + 1601 (with n = 80);

• the smallest solution of γ(n + 1) − γ(n) = 17: the only solutions n

109

are

1 681, 59 535, 139 239 and 505 925 (see the footnote tied to the number 98);

• the second number n divisible by a square 1 and such that δ(n+1)−δ(n) = 1

(see the number 49);

• the fourth solution w + s of the aligned houses problem (see the number 35).

1 682

• the fifth number n such that the binomial coeﬃcient

(

n

2

)

is a perfect square:

here

(1682)

2

=

11892

(see the number 289).

1 701 (=

35

· 7)

• the second number n divisible by a square 1 and such that γ(n+14) = γ(n)+

14: the smallest is 49 (and there are no others 108); here γ(n + 14) − γ(n) =

γ(1715) − γ(1701) = γ(73 · 5) − γ(35 · 7) = 35 − 21 = 14.