Those Fascinating Numbers 137
1 849
the smallest solution of τ (n + 1) τ (n) = 9; the sequence of numbers satisfying
this equation begins as follows: 1849, 11449, 23103, 28899, 38415, 63001, 66049,
195363, . . . (see the number 399).
1 851
the smallest number n such that P (n) P (n + 1) . . . P (n + 5): here
617 463 109 103 53 29; if we denote by nk the smallest number n
such that P (n) P (n + 1) . . . P (n + k 1), then we have the following
table:
k 2 3 4 5 6 7 8
nk 5 13 13 491 1 851 12 721 12 721
k 9 10 11 12 13
nk 109 453 586 951 120 797 465 624 141 002 4 044 619 541
(compare with the table displayed with the number 46 189, this time for the
sequence of increasing P (n + i)).
1 853 (= 17 · 109)
the number n which allows the sum
m≤n
ω(m)=2
1
m
to exceed 3 (see the number 44).
1 854
the number of possible arrangements of the integers 1,2,. . . ,7 with the restric-
tion that the integer j must not be in the j-th position for each j, 1 j 7
(see the number 265).
1 867
the smallest prime number p such that ω(p+1) = 2, ω(p+2) = 3 and ω(p+3) =
4 (see the number 103).
1 873
the tenth prime number pk such that p1p2 . . . pk 1 is prime (see the number
317).
1 876
the rank of the prime number which appears the most often as the
17th
prime
factor of an integer: p1876 = 16 111 (see the number 199).
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