Those Fascinating Numbers 151
2 730
the denominator of the Bernoulli number B12 =
691
2730
.
2 737
the smallest number n such that ι(n) = ι(n + 1) = . . . = ι(n + 6) = 1; here ι(n)
stands for the index of isolation of n and is defined by ι(n) = min
1≤m=n
P (m)≤P (n)
|n m|,
that is the distance to the nearest integer whose largest prime factor is no
larger than that of n; if nk stands for the smallest number n such that ι(n) =
ι(n + 1) = . . . = ι(n + k 1) = 1, then we have the following table:
k 6 7 8 9 10 11
nk 169 2 737 26 536 67 311 535 591 3 021 151
k 12 13 14 15
nk 26 817 437 74 877 777 657 240 658 785 211 337
2 749
the smallest number n for which the Moebius function µ takes successively,
starting with n, the values −1, 0, −1, 0, −1, 0, −1, 0; if nk stands for the smallest
number n for which the Moebius function µ takes successively, starting with n,
the values −1, 0, −1, 0, . . . , −1
or
0,
k
numbers
then n2 = n3 = 3, n4 = 17, n5 = n6 = 195,
n7 = 1 547, n8 = 2 749, n9 = n10 = n11 = 4 011, n12 = 462 649, n13 = n14 =
n15 = 580 547 and n16 = 74 406 035: one can easily prove that there are
no142
numbers nk with k 17 (for the sequence 1, 0, 1, 0, . . ., see the number 3 647).
2 821 (= 7 · 13 · 31)
the fifth Carmichael number (see the number 561).
2 834
the
13th
solution of φ(n) = φ(n + 1) (see the number 15).
2 835
the fourth odd abundant number (see the number 945).
142This easily follows from the fact that in any sequence of 17 consecutive numbers, the first of
which is square-free, there must be two numbers r and s both divisible by 9 and thus such that
µ(r) = µ(s) = 0 with r s odd.
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