60 Jean-Marie De Koninck

• the smallest number n such that 6! divides 1 + 2 + . . . + n: if we denote by nk

the smallest number such that k! divides 1 + 2 + . . . + nk, then n2 = n3 = 3,

n4 = n5 = 15, n6 = n7 = 224, n8 = 4 095, n9 = 76 544, n10 = 512 000,

n11 = 9 511 424 and n12 = 20 916 224;

• the second solution of equation

σ(n)

n

=

9

4

(see the number 40);

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

(see the number 48);

• the seventh Granville number (see the number 126).

225

• the largest solution x of the diophantine equation

x2

+ 28 =

y3

(W.J. Ellison,

F. Ellison, J. Pesek, C.E. Stahl, D.S. Stall [75]): the only solutions (x, y) of this

diophantine equation are (6, 4), (22, 8) and (225, 37); a diophantine equation

of the form

x2

+ k =

y3,

where k is a positive integer, is sometimes called a

Bachet equation (see the number 251).

227

• the smallest number n which allows the sum

i≤n

1

i

to exceed 6 (see the number

83);

• the third Stern number (see the number 137).

228

• the smallest number n ≥ 2 such that n|σ48(n); the sequence of numbers sa-

tisfying this property begins as follows: 228, 386, 444, 876, 1308, 2812, 5196,

5548, 6924, 7372, 8284, 8436, . . .

229

• the second prime number p such that 44p−1 ≡ 1 (mod p2): the only prime

numbers p 232 satisfying this congruence are 3, 229 and 5 851 (see Ribenboim

[169], p. 347).

230

• the smallest number n such that ω(n) = ω(n + 1) = 3: here 230 = 2 · 5 · 23 and

231 = 3 · 7 · 11; if nk stands for the smallest number n such that n and n + 1

each have k distinct prime factors (that is such that ω(n) = ω(n + 1) = k),

then n1 = 2, n2 = 14, n3 = 230, n4 = 7 314, n5 = 254 540, n6 = 11 243 154 and