Those Fascinating Numbers 31
begins as follows: 1, 6, 92, 406, 1 549, 5 361, 12 546, 41 908, 141 121, . . . ; it is
well known that the function M(x) changes sign infinitely often as x (see
E. Grosswald [98]), which implies in particular that the number mk does indeed
exist for each k 1.
93
the largest known number k such that the decimal expansion of 2k does not
contain the digit 6; indeed, using a computer, one can verify that
293
= 9903520314283042199192993792
does not contain the digit 6, while each number
2k,
for k = 94, 95, . . . , 3000,
contains it (see the number 71);
the smallest number 1 whose sum of divisors is a seventh power: σ(93) =
27.
94
the seventh number whose sum of divisors is a perfect square: σ(94) = 112 the
sequence of numbers satisfying this property begins as follows: 1, 3, 22, 66, 70,
81, 94, 115, 119, 170, 210, 214, . . .
96
the smallest number n such that each number m n can be written as the
sum of distinct elements from the set {ppn : n = 1, 2, . . .} = {3, 5, 11, 17, . . .}
(see R.E. Dressler & S.T. Packer [69]);
the smallest number m such that equation σ(x) = m has exactly four solutions,
namely 42, 62, 69 and 77;
the smallest number n 1 such that
γ(n)2|σ(n):
the sequence of numbers satis-
fying this property begins as follows: 1, 96, 864, 1080, 1782, 6144, 7128, . . .
36;
if nk stands
for37
the smallest number n for which
σ(n)/γ(n)k
is an integer,
then n1 = 6, n2 = 96, n3 = 3 538 944 and n4 19 698 744 770 118 549 504 (see
the number 1 782).
97
the smallest prime number preceded by exactly seven consecutive composite
numbers; indeed, there are no prime numbers between 89 and 97; if qk stands for
the smallest prime number that is preceded by exactly k consecutive composite
numbers, then q1 = 5, q3 = 11, q5 = 29, q7 = 97, q9 = 149, q11 = 211,
q13 = 127, q15 = 1 847, q17 = 541, q19 = 907 and q21 = 1 151;
the largest two digit prime number.
36It
is easy to see that there exist infinitely many numbers satisfying this property, namely all
the numbers n =
2α3β
, where α + 1 is a multiple of 6 and β 1 is odd.
37By
examining the numbers of the form n =
2α3β
with an appropriate choice of α and β, one
can easily discover infinitely many numbers n for which γ(n)k|σ(n) for any fixed number k.
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