Those Fascinating Numbers 157
3 605
the number of Carmichael numbers 1011 (see the number 646).
3 610
the
20th
number n such that n! 1 is prime (see the number 166).
3 647
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 we denote by nk 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 we have n2 = 15, n3 = 55, n4 = 159,
n5 = n6 = n7 = 411, n8 = 3 647, n9 = n10 = n11 = 15 243, n12 = 113 343,
n13 = n14 = n15 = 1 133 759 and n16 = 29 149 139: one can easily prove that
there are no146 numbers nk with k 17 (for the sequence −1, 0, −1, 0, . . ., see
the number 2 749).
3 655
the second number n such that n, n + 1, n + 2 and n + 3 have the same number
of divisors, namely eight (see the number 242).
3 675 (= 3 ·
52
·
72)
the smallest odd number n having exactly three prime factors and such that
γ(n)|φ(n).
3 684
the 13th Keith number (see the number 197).
3 705
the
15th
solution of φ(n) = φ(n + 1) (see the number 15).
3 759
the seventh number such that
2n
+
n2
is prime (see the number 2 007).
146This simply follows from the fact that any sequence of 17 consecutive numbers, the first of
which is square-free, must include two numbers r and s both divisible by 9 and therefore such that
µ(r) = µ(s) = 0 with r s odd.
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