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).