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.
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