Those Fascinating Numbers 103

719

• the smallest prime factor of the Mersenne number

2359

− 1, whose complete

factorization is given by

2359

− 1 = 719 · 855857 · 778165529 · 65877330027880703

·370906580744492785430299503112990447

·100361196281293745682520861860411315001;

• the smallest prime number of height 6 (see the number 283).

720 (=

24

·

32

· 5)

• the

14th

highly composite number (see the number 180);

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

726

• the first of the six smallest consecutive numbers at which the Ω(n) function

takes distinct values, namely here the values 4, 1, 5, 6, 3 and 2: if nk, k ≥ 2,

stands

for107

the first of the k smallest consecutive numbers at which the Ω(n)

function takes k distinct values, then n2 = 3, n3 = 6, n4 = 15, n5 = 60,

n6 = 726, n7 = 6 318, n8 = 189 375, n9 = 755 968 and n10 = 683 441 871 (see

the number 417 for the same matter, but this time with the ω(n) function).

727

• the smallest prime number p such that p + 2 is108 a cube: here 727 + 2 = 93;

the sequence of numbers satisfying this property begins as follows: 727, 3373,

6857, 19681, 29789, 50651, 300761, 753569, 970297, 1295027, . . . ;

• the third prime number of the form n!+ n +1, here with n = 6 (see the number

52).

729

• the fifth solution of φ(σ(n)) = σ(φ(n)) (see the number 516).

732

• the tenth number n for which the distance from en to the nearest integer is the

smallest (see the numbers 178 and 58).

107See the footnote tied to the number 417.

108It

would be interesting if one could prove that this sequence is infinite. This could be a diﬃcult

task since on the one hand it is not known if the sequence (n3 − 2)n≥3 contains infinitely many

prime numbers, while on the other hand the challenge could be similar to that of the twin prime

conjecture.