Those Fascinating Numbers 287

3 653 786

• the second number which is equal to the sum of the seventh powers of its digits

added to the product of its digits (see the number 455 226).

3 678 723

• the smallest number n such that P (n + i) ≤

3

√

n + i, for i = 0, 1, 2, 3: here

P (3678723) = P

(33

· 19 · 71 · 101) = 101 3

√

3678723 ≈ 154, P (3678724) =

P

(22·72·1372)

= 137 3

√

3678724 ≈ 154, P (3678725) = P

(52·37·41·97)

= 97

3

√

3678725 ≈ 154 and P (3678726) = P (2 · 3 ·

832

· 89) = 89 3

√

3678726 ≈ 154

(see the number 134 848).

3 741 154 (= 2 · 1 870 577)

• the 1 000

000th

number having exactly two distinct prime factors (see the num-

ber 184).

3 768 373

• the third number n such that Eσ(n) := σ(n + 1) − σ(n) satisfies Eσ(n + 1) =

Eσ(n): here the common value of Eσ is 1 100 736, since σ(3768373) = 4560192,

σ(3768374) = 5660928 and σ(3768375) = 6761664 (see the number 693).

3 847 271

• the smallest prime number p such that ω(p−1) = ω(p+1) = 6: here 3 847 270 =

2 · 5 · 7 · 17 · 53 · 61 and 3 847 272 = 23 · 3 · 11 · 13 · 19 · 59 (see the number 131).

3 885 569

• the smallest prime number p such that Ω(p − 1) = Ω(p + 1) = 10: here

3 885 568 =

29

· 7589 and 3 885 570 = 2 ·

36

· 5 · 13 · 41 (see the number 271).

3 995 648

• the smallest number n such that n and n + 1 are both divisible by a ninth

power: 3 995 648 =

211

· 1951 and 3 995 649 =

39

· 7 · 29 (see the number 1 215).

4 037 913

• the value of 1! + 2! + . . . + 10!.

4 046 849

• the largest known prime number p such that P (p2 −1) ≤ 23 and P (p2 −1) = 23:

here

p2

− 1 =

215

·

32

·

52

· 13 · 17 · 19 ·

232

(see the number 4 801).