68 Jean-Marie De Koninck

279

• the smallest number r which has the property that each number can be written

in the form x1

8

+ x2

8

+ . . . + xr,

8

where the xi’s are non negative integers (see the

number 4).

280

• the third solution of σ2(n) = σ2(n + 2) (see the number 1 079).

283

• the prime number which appears the most often as the ninth prime factor of

an integer (see the number 199);

• the smallest prime number of height 5; we say that a prime number p is of height

r if the number of required iterations to reach the prime number 2 in the process

P (p − 1) = q, p = q, P (p − 1) = q, . . . , is r; thus the height of 283 is 5 because

P (283 − 1) = 47, P (47 − 1) = 23, P (23 − 1) = 11, P (11 − 1) = 5, P (5 − 1) = 2;

if qk stands for the smallest prime number of height k, then q2 = 7, q3 = 23,

q4 = 47, q5 = 283, q6 = 719, q7 = 1 439, q8 = 2 879, q9 = 34 549, q10 = 138 197,

q11 = 1 266 767 and q12 = 14 619 833.

284

• the number which, when paired with the number 220, forms the smallest pair

of amicable numbers: σ(220) = σ(284) = 504.

287

• the fourth number n such that σ2(n) is a perfect square: here σ2(287) =

2902

(see the number 42).

where B = 0.261497212847643 . . . (see J.B. Rosser & L. Schoenfeld [178]), we obtain that

1.7799 ·

1018

q4 1.8228 ·

1018.

If the Riemann Hypothesis is true, then

log log q4 + B −

3 log q4 + 4

8π

√

q4

p≤q4

1

p

log log q4 + B +

3 log q4 + 4

8π

√

q4

(see L. Schoenfeld [182]), in which case we obtain the sharper bounds

1.8012409 ·

1018

q4 1.8012416 ·

1018.