Those Fascinating Numbers 29

number of the narcissistic numbers

digits

32 17333509997782249308725103962772

33 186709961001538790100634132976990,

186709961001538790100634132976991

34 1122763285329372541592822900204593

35 12639369517103790328947807201478392,

12679937780272278566303885594196922

37 1219167219625434121569735803609966019

38 12815792078366059955099770545296129367

39 115132219018763992565095597973971522400,

115132219018763992565095597973971522401

• the smallest solution of σ(n) = σ(n + 30): the sequence of numbers satisfying

this equation begins as follows: 88, 161, 164, 209, 221, 275, 279, 376, 497, 581,

707, 869, 910, 913, . . .

89

• the exponent of the tenth Mersenne prime

289

− 1 (Powers, 1911);

• the smallest prime number amongst those which appear more often as the third

prime factor of an integer than as the second prime factor (in this case, when

89 appears in the factorization of a number, it is the third prime factor of that

number in 31.6% of the cases, while it is the second one in 27.9% of the cases,

the fourth in 17% of the cases, the fifth one in 6% of the cases and the sixth

one in only 1% of the cases): this result can be obtained from those published

in a paper by J.M. De Koninck & G. Tenenbaum [63];

• the only solution Fn (where Fn stands for the

nth

Fibonacci number)

of35

1

Fn

=

∞

k=0

Fk

10k+1

;

see B.B.M. de Weger [200];

• the smallest number n 9 satisfying n = d1 + d2

2

+ d3

4

+ . . . +

drr−1

2

, where

d1, d2, . . . , dr stand for the digits of n; the only other known number satisfying

this property is 6 603;

• the fifth prime Fibonacci number: the sequence of such numbers begins as

follows 2, 3, 5, 13, 89, 233, 1 597, 28 657, 514 229, 433 494 437, 2 971 215 073,

99 194 853 094 755 497, 1 066 340 417 491 710 595 814 572 169,

35This

result can be compared with that of Cohn (see the number 144) according to which

144 is the only Fibonacci number which is a perfect square. Indeed, it is easy to establish that

∞

k=0

Fk

mk+1

=

1

m2

− m − 1

for each m ≥ 2. Thus, the property

1

Fn

=

∞

k=0

Fk

10k+1

is equivalent to

the relation Fn = m2 − m − 1, whose only solution is m = 10, Fn = 89.