Those Fascinating Numbers 25
77
the largest number which cannot be written as the sum of positive integers
whose sum of reciprocals is equal to 1; thus
78 = 2 + 6 + 8 + 10 + 12 + 40 and
1
2
+
1
6
+
1
8
+
1
10
+
1
12
+
1
40
= 1
(R.L. Graham [95]);
the smallest champion number whose smallest prime factor is 7: we say that n
is a champion number if D(n) D(m) for each positive integer m n, where
D(n) = τ (n!) τ ((n 1)!) (see A. Ivi´ c & C. Pomerance [112]).
78
the smallest solution of σ2(n) = σ2(n + 13);
the number of digits in the decimal expansion of the Fermat number
228
+ 1;
the number of pseudoprime numbers in base 2 smaller than 107; we say that
an odd composite number n is a pseudoprime number in base 2 if 2n−1 1
(mod n); see the number 245;
the largest known number k such that the decimal expansion of 2k does not
contain the digit 8; indeed, using a computer, one can verify that
278
= 302231454903657293676544
does not contain the digit 8, while each number
2k,
for k = 79, 80, . . . , 3000,
contains it (see the number 71);
79
the smallest number which cannot be written as the sum of less than 19 fourth
powers: here 79 = 15 · 14 + 4 · 24;
the second prime number p such that 31p−1 1 (mod p2): the only prime
numbers p 232 satisfying this congruence are 7, 79, 6 451 and 2 806 861 (see
Ribenboim [169], p. 347).
80
the smallest number n such that τ (n) τ (n + 1) τ (n + 2) τ (n + 3): here
10 5 4 2 (see the number 45);
the second number for which the product of its proper divisors is a fourth
power, that is such that
d|n, dn
d =
a4:
here
2 · 4 · 5 · 8 · 10 · 16 · 20 · 40 = 40 960 000 =
804;
48 is the smallest number satisfying this property.
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