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).
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