Those Fascinating Numbers 265

614 341

• the smallest number n 1 for which ξ(n) is an integer, where ξ(n) stands for

n

i=1

1

gcd(i, n)

: here183 ξ(n) = 486 361; the sequence of numbers satisfying this

property begins as follows: 614 341, 618 233, 1 854 699, 11 746 427, 26 584 019,

35 239 281, 79 752 057, 85 393 399, 118 082 503, 345 592 247, 354 247 509,

505 096 361, 802 597 537, 1 036 776 741, 1 062 742 527, 1 515 289 083, 2 149 579 159,

2 243 567 557, 3 695 178 641, 5 077 547 629, . . .

184

614 656 (=

284)

• the fourth number n 1 whose sum of digits is equal to

4

√

n (see the number

2 401).

614 657

• the eighth prime number of the form

n4

+ 1, here with n = 28 (see the number

1 297).

617 057

• the smallest number which can be written as the sum of two and three distinct

fourth powers: 617 057 =

74

+

284

=

34

+

204

+

264;

the sequence of num-

bers satisfying this property begins as follows: 617 057, 1 957 682, 3 502 322,

3 959 297, 6 959 682, 9 872 912, 31 322 912, 40 127 377, 46 712 801, 48 355 137,

49 981 617,. . . (see the number 4 802).

617 139

• the smallest number n such that β(n) ≤ β(n + 1) ≤ . . . ≤ β(n + 8): here

423 589 811 1487 1616 7026 41151 308575 617147 (see the

number 714).

618 233

• the second number n for which ξ(n) is an integer (see the number 614 341).

627 208 (= 23 · 78401)

• the number n which allows the sum

m≤n

ω(m)=2

1

m

to exceed 5 (see the number 44).

183One

can easily show that ξ(n) =

1

n

pa

n

p2a+1+1

p+1

.

184It would be interesting if one could show that this sequence is infinite.