Those Fascinating Numbers 121

1 187

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

1 189

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

1 197

• the 1

000th

composite number (see the number 133).

1 201

• the fourth prime number of the form

(x4

+

y4)/2:

here 1 201 =

(14

+

74)/2

(see

the number 41).

1 210

• the number which, when paired with the number 1 184, forms an amicable pair

(see the number 1 184).

1 215

• the smallest number n such that n and n +1 are both divisible by a fifth power:

1215 = 35 · 5, 1216 = 26 · 19; if nk stands for the smallest number n such that n

and n + 1 are both divisible by a kth power, then n2 = 8, n3 = n4 = 80, n5 =

1 215, n6 = 16 767, n7 = 76 544, n8 = 636 416, n9 = 3 995 648, n10 = 24 151 040

and n11 = 36 315 135 (for the analogue problem concerning three consecutive

integers, see the number 1 375).

1 218 (= 2 · 3 · 7 · 29)

• the smallest solution of

φ(n)

n

=

8

29

; the sequence of numbers satisfying this

equation begins as follows: 1218, 2436, 3654, 4872, 7308, 8526, 9744, . . . ; it is

easy to establish that this sequence is infinite123.

123Indeed,

this follows from the fact that one can easily prove that n is a solution of this equation

if and only if n = 2α · 3β · 7γ · 29δ for certain positive integers α, β, γ, δ.