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).
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