66 Jean-Marie De Koninck

• the second number n such that n2 + 2 is powerful: here 2652 + 2 = 70 227 =

35 · 172; the six smallest numbers satisfying this property are 5, 265, 13 775,

716 035, 9 980 583 and 37 220 045.73

269

• the smallest prime number q such that

∑

p≤q

p is divisible by 6 (= 2 · 3): here

this sum is equal to 6 870 and it is also divisible by 30 (= 2 · 3 · 5); if we

denote by qk the smallest prime number such that

p≤pk

p divides

∑

p≤qk

p,

then q2 = q3 = 269, q4 = 3 823, q5 = 8 539, q6 = 729 551, q7 = 1 416 329,

q8 = 23 592 593 and q9 = 1 478 674 861.

270

• the smallest number n such that (σI (n) + γ(n))/n is an integer; the only num-

bers n

108

satisfying this property are n = 270, 87 750 and 217 854; observe

that in each of these cases, (σI (n) + γ(n))/n = 3;

• the third number which is neither perfect or multi-perfect but whose harmonic

mean is an integer (see the number 140);

• the second solution of

σ(n)

n

=

8

3

(see the number 1 488).

271

• the smallest prime number p such that Ω(p − 1) = Ω(p + 1) = 5: here 270 =

2·33

·5 and 272 =

24

·17; if we denote by p(k) the smallest prime number p such

that Ω(p − 1) = Ω(p + 1) = k, then p(2) = 2, p(3) = 19, p(4) = 89, p(5) = 271,

p(6) = 1 889, p(7) = 10 529, p(8) = 75 329, p(9) = 157 951, p(10) = 3 885 569,

p(11) = 11 350 529, p(12) = 98 690 561 and p(13) = 169 674 751;

• the second prime number built from the first digits of the decimal expansion of

the Euler constant e = 2.718281828 . . .: the sequence of these prime numbers

begins as follows: 2, 271, 2 718 281, . . . (the fourth has 84 digits);

• the smallest prime number p which is the first link of a

p2

+1 chain of order 4; we

say that a prime number q1 ≥ 3 is the first link of a

p2

+1 chain of order k ≥ 3 if

the numbers q1, q2 = (q1

2

+1)/2, . . . , qk = (qk−1

2

+1)/2 are all primes; thus, here

q1 = 271, q2 =

(2712

+ 1)/2 = 36 721, q3 =

(367212

+ 1)/2 = 674 215 921 and

q4 =

(6742159212

+ 1)/2 = 227 283 554 064 939 121 are all primes; the smallest

prime numbers p which are the first links of a p2 + 1 chain of order k, for

k = 2, 3, 4, 5, are respectively 3, 3, 271, 169 219;

• the largest prime factor of 123454321, whose complete factorization is given by

123454321 = 412 · 2712.

73Such a number n must be odd since otherwise 2 (n2 + 2), in which case n2 + 2 is not powerful.

On the other hand, it is important to mention that there exist infinitely many numbers satisfying

this property (see the footnote tied to the number 37).