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


q4
p≤q4
1
p
log log q4 + B +
3 log q4 + 4


q4
(see L. Schoenfeld [182]), in which case we obtain the sharper bounds
1.8012409 ·
1018
q4 1.8012416 ·
1018.
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