Those Fascinating Numbers 213

37 960

• the smallest number n such that ω(n) = ω(n + 1) = ω(n + 2) = 4; the sequence

of numbers satisfying this property begins as follows: 37 960, 44 484, 45 694,

50 140, 51 428, 55 130, 55 384, 61 334,. . . ; if nk, stands for the smallest number

n such that

ω(n) = ω(n + 1) = . . . = ω(n + k − 1) = ,

here are the values of some of the nk,:

(a star next to a number indicates that it is “most likely” the smallest with

that property)

ω(n) = k = 2 k = 3 k = 4

= 2 14 20 33

= 3 230 644 1 308

= 4 7 314 37 960 134 043

= 5 254 540 1 042 404 21 871 365

= 6 11 243 154 323 567 034 10 744 118 592(*)

= 7 965 009 045 88 533 549 754(*) 29 694 692 454(*)

ω(n) = k = 5 k = 6 k = 7

= 2 54 91 141

= 3 2 664 6 850 10 280

= 4 357 642 1 217 250 1 217 250

= 5 129 963 314 830 692 265 4 617 927 894

ω(n) = k = 8 k = 9 k = 10 k = 11

= 2 141 nil nil nil

= 3 39 693 44 360 48 919 218 972

= 4 14 273 478 44 939 642 70 067 298 163 459 752

ω(n) = k = 12 k = 13 k = 14

= 2 nil nil nil

= 3 526 095 526 095 526 095

= 4 547 163 235 2 081 479 430 2 771 263 512

(see the number 242 for the analogue problem with the function Ω(n)).

39 325

• the fifth number n divisible by a square 1 and such that δ(n + 1) − δ(n) = 1

(see the number 49).

39 607

• the first term of the smallest sequence of nine consecutive prime numbers all

of the form 4n + 3 (see the number 463).