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
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