240 Jean-Marie De Koninck
132 049
the exponent of the
30th
Mersenne prime
2132 049
1 (Slowinski, 1983).
133 496
the number of possible arrangements of the integers 1,2,. . . ,9 with the restric-
tion that the jth integer must not be in the jth position for each j, 1 j 9
(see the number 265).
134 043
the smallest number n such that ω(n) + ω(n + 1) + ω(n + 2) + ω(n + 3) = 16:
here 134 043 = 3 · 7 · 13 · 491, 134 044 =
22
· 23 · 31 · 47, 134 045 = 5 · 17 · 19 · 83
and 134 046 = 2 ·
32
· 11 · 677 (see the number 987).
134 848
the smallest number n such that P (n + i) ≤√3

n + i, for i = 0, 1, 2: here
P (134848) = P (26 · 72 · 43) = 43 51.28 . . . =
3
134848, P (134849) = P (11 ·
13 · 23 · 41) = 41 51.28 . . . =
3

134849 and P (134850) = P (2 · 3 · 52 · 29 · 31) =
31 51.28 . . . = 3

134850; if nk stands for the smallest number n such that
P (n + i)
3

n + i, for i = 0, 1, 2, . . . , k 1, then n2 = 2 400, n3 = 134 848 and
n4 = 3 678 723 (see the numbers 1 518 and 290 783).
138 125 (=
54
· 13 · 17)
the smallest number which can be written as the sum of two squares in ten
distinct ways, namely 138 125 = 222 + 3712 = 352 + 3702 = 702 + 3652 =
1102 + 3552 = 1252 + 3502 = 1632 + 3342 = 1942 + 3172 = 2052 + 3102 =
2182 + 3012 = 2502 + 2752 (see the number 50).
138 240
the fourth number which is equal to the product of the factorials of its digits
in base 6:
138 240 = [2, 5, 4, 4, 0, 0, 0]6 = 2! · 5! · 4! · 4! · 0! · 0! · 0!;
the only known numbers satisfying this property are 1, 2, 24 and 138 240.
139 239
the third solution of γ(n + 1) γ(n) = 17 (see the number 1 681).
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