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.