Those Fascinating Numbers 15
39
the smallest number n such that 2n 7 is prime (a question raised by Erd˝os
in 1956): using a computer, one obtains that the other numbers n 20 000
satisfying this property are 715, 1983, 2319, 2499, 3775 and 12819.19
40
the smallest solution of equation
σ(n)
n
=
9
4
; the sequence of numbers satisfying
this equation begins as follows: 40, 224, 174 592, 492 101 632, . . .
41
the largest odd number which is not the sum of four non zero squares (Sierpinski
[185], p. 404);
the largest number n such that the polynomial
x2
+ x + n is prime for each of
the numbers x = 0, 1, 2, . . . , n 2; the other numbers n satisfying this property
are n = 1, 2, 3, 5, 11 and 17 (see D. Fendel & R.A. Mollin [80]);
the integer part of the number γ0 = 41.677647, that is the conjectured value
of lim supn→∞
σ∞(n)
log n
, where σ∞(n) stands for the smallest number k such that
f
k(n)
= 1, where
f(n) =



1 if n = 1,
n/2 if n is even,
3n + 1 if n is odd,
f
1(n)
= f(n), f
2(n)
= f(f(n)), f
3(n)
= f(f
2(n))
and so on; according to the
Syracuse conjecture (also called the 3x + 1 problem ), this sequence inevitably
reaches the number 1 (see J.C. Lagarias & A. Weiss [120]);
the smallest prime number of the form (x4 +y4)/2: here 41 = (34 +14)/2: there
exist only eight prime numbers 10 000 satisfying20 this property, namely 41,
313, 353, 1201, 3593, 4481, 7321 and 8521.
42
the smallest number n 1 such that σ2(n) is a perfect square: the sequence
of numbers satisfying this property begins as follows: 42, 246, 287, 728, 1434,
1673, 1880, 4264, 6237, 9799, 9855,. . . ;
the fifth Catalan number (see the number 14);
19While
performing this search, one can ignore all even numbers n, all numbers n 1 (mod 4),
all numbers n 7 (mod 10) as well as all numbers n 11 (mod 12), since in these four cases,
we obtain respectively that 3, 5, 11 and 13 divide 2n 7.
20It
is interesting to mention that there exist other forms which generate infinitely many prime
numbers: it is the case, for instance, for the form x2 + y4 as it was proved by J. Friedlander &
H. Iwaniec [85], as well as for the form
x3
+
2y3
as was established by D.R. Heath-Brown [110] (see
also for that matter the number 3 391).
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