96 Jean-Marie De Koninck

607

• the exponent of the 14th Mersenne prime 2607 − 1 (Robinson, 1952).

608

• the fifth number n such that the corresponding triangular number Tn =

n(n+1)

2

is the product of three consecutive numbers: here

608 · 609

2

= 56 · 57 · 58; there

exist only six triangular numbers satisfying this property, namely T3, T15, T20,

T44, T608 and T22736 (N. Tzanahis & B.M.M. de Weger [197]).

609

• the smallest solution of

φ(n)

n

=

16

29

; the sequence of numbers satisfying this

equation begins as follows: 609, 1827, 4263, 5481, 12789, 16443, 17661, 29841,. . .

610

• the

12th

Markoff number (see the number 433).

613

• the rank of the 23rd Lucas prime number: a Lucas number is a member of the

recurrence sequence (

n

)n≥1 defined by

1

= 1,

2

= 3 and

n+1

=

n

+

n−1

for each integer n ≥ 2; the prime numbers belonging to this sequence are called

Lucas prime numbers; the only known numbers n such that

n

is prime are: 2,

4, 5, 7, 8, 11, 13, 16, 17, 19, 31, 37, 41, 47, 53, 61, 71, 79, 113, 313, 353, 503,

613, 617, 863, 1097, 1361, 4787, 4793, 5851, 7741, 8467, 10691, 12251, 13963,

14449, 19469, 35449, 36779, 44507 and 51169.

616

• the smallest number n which allows the sum

i≤n

1

i

to exceed 7 (see the number

83).

617

• the number of digits in the decimal expansion of the Fermat number

2211

+ 1.