224 Jean-Marie De Koninck
65 535
the fifth number n such that φ(n)|(n+1); the seven smallest numbers satisfying
this property are 1, 3, 15 (= 3 · 5), 255 (= 3 · 5 · 17), 65 535 (= 3 · 5 · 17 · 257),
83 623 935 (= 3 · 5 · 17 · 353 · 929) and 4 294 967 295 (= 3 · 5 · 17 · 257 · 65537);
the number 6 992 962 672 132 095 = 3 · 5 · 17 · 353 · 929 · 83 623 937 also satisfies
this property (see R.K. Guy [101], B37);
the ninth number n such that σ(φ(n)) = n (see the number 744).
65 537
the fifth Fermat number: 65 537 =
224
+ 1; it is the largest known Fermat
prime; the other known Fermat primes are 3, 5, 17 and 257.
66 161
the smallest prime number p such that 6p−1 1 (mod p2): the only prime
numbers p 232 satisfying this congruence are 66 161, 534 851 and 3 152 573.
66 198
the fourth Giuga number (see the number 30).
67 187
the third prime number q such that

p≤q
p is a perfect square: here

p≤67187
p = 212372329 = 145732 (see the number 22 073).
68 832
the ninth number n such that σ(n) and σ2(n) have the same prime factors,
namely the primes 2, 3, 5, 7 and 13 (see the number 180).
68 889
the smallest number of persistence 7 (see the number 679).
68 906
the number of six digit prime numbers (see the number 21).
69 623
the smallest prime number equally distant, by a distance of 30, from the
preceding and following prime numbers: p6905 = 69 593, p6906 = 69 623 and
p6907 = 69 653.
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