Those Fascinating Numbers 241

141 121

• the number of integer zeros of the function M(x) :=

n≤x

µ(n) located in the

interval [1,

109]

(see the number 92).

142 560

• the ninth number n 1 such that φ(σ(n)) = n (see the number 128).

142 857

• the smallest number which quintuples when its last digit is moved in first po-

sition173;

• the first six decimals (and the period) of 1/7; indeed,

1

7

= 0.142857 142857 . . .

143 018

• the

25th

number n such that n ·

2n

− 1 is prime (see the number 115).

145 823

• the 20th prime number pk such that p1p2 . . . pk + 1 is prime (see the number

379).

147 492

• the sixth number n such that φ(n) + σ(n) = 3n (see the number 312).

148 219 (= 19 · 29 · 269)

• the smallest square-free composite number n such that p|n =⇒ p + 6|n + 6 (see

the number 399).

148 349

• the only known number n which is equal to the sum of the sub-factorials of its

digits: 148 349 =!1+!4+!8+!3+!4+!9; the sub-factorial function !n is defined by

!n = n! 1 −

1

1!

+

1

2!

−

1

3!

+ . . . +

(−1)n

1

n!

.

173One

can prove that there exist infinitely many numbers satisfying this property by simply

considering the sequence of numbers 142 857, 142 857 142 857, 142 857 142 857 142 857, and so on.