Those Fascinating Numbers 215
40 585
the largest number which is equal to the sum of the factorials of its digits (see
the number
145).162
40 625
most likely the number z at which the expression
1
λ(x2)
+
1
λ(y2)
+
1
λ(z2)
, where
x2 + y2 = z2, reaches its minimal value: here x = 35 · 149, y = 23 · 72 · 47,
z = 55 · 13, and
1
λ(x2)
+
1
λ(y2)
+
1
λ(z2)
0.817721.
40 723
the smallest number n such that φ(n) = 8! (see the number 779).
40 755
the second number 1 which is both triangular and pentagonal: 40 755 =
285·286
2
=
165(3·165−1)
2
; the smallest number satisfying this property is 210.
41 041 (= 7 · 11 · 13 · 41)
the smallest Carmichael number which is the product of four prime numbers:
it is the
11th
Carmichael number; if we denote by nk the smallest Carmichael
number having exactly k prime factors, we have the following table:
k nk
3 561 = 3 · 11 · 17
4 41 041 = 7 · 11 · 13 · 41
5 825 265 = 5 · 7 · 17 · 19 · 73
6 321 197 185 = 5 · 19 · 23 · 29 · 37 · 137
7 5 394 826 801 = 7 · 13 · 17 · 23 · 31 · 67 · 73
8 232 250 619 601 = 7 · 11 · 13 · 17 · 31 · 37 · 73 · 163
9 9 746 347 772 161 = 7 · 11 · 13 · 17 · 19 · 31 · 37 · 41 · 641
10 1 436 697 831 295 441 = 11 · 13 · 19 · 29 · 31 · 37 · 41 · 43 · 71 · 127
11 60 977 817 398 996 785 = 5 · 7 · 17 · 19 · 23 · 37 · 53 · 73 · 79 · 89 · 233
12 7 156 857 700 403 137 441
= 11 · 13 · 17 · 19 · 29 · 37 · 41 · 43 · 61 · 97 · 109 · 127
13 1 791 562 810 662 585 767 521
= 11 · 13 · 17 · 19 · 31 · 37 · 43 · 71 · 73 · 97 · 109 · 113 · 127
14 87 674 969 936 234 821 377 601
= 7 · 13 · 17 · 19 · 23 · 31 · 37 · 41 · 61 · 67 · 89 · 163 · 193 · 241
162It is not known if there exists a number 2 which is equal to the product of the factorials of
its digits in base 10. However, some numbers are equal to the product of their digits in other bases
(see the number 17 280).
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