316 Jean-Marie De Koninck

71 265 503

• the second number n such that φ(n), φ(n + 1), φ(n + 2), φ(n + 3), φ(n + 4) and

φ(n + 5) have the same prime factors, namely here the primes 2, 3, 5, 7 and

101 (see the number 43 570 803).

71 315 748

• the eighth number n such that σ3(n) is a perfect square: indeed,

σ3(71315748) =

6564899577602

(see the number 345).

73 939 133

• the largest known right truncatable prime number: such a number has the

property that, regardless the number of digits we remove from the right, the

remaining number is still prime: thus 7393913, 739391, 73939, 7393, 739,

73 and 7 are all primes; such a number is also sometimes called a super-

prime number (see I.O. Angell & H.J. Godwin [4], and see also the number

357 686 312 646 216 567 629 137 for the largest known left truncatable prime

number).

75 007 400

• the

19th

number n such that φ(n) + σ(n) = 3n (see the number 312).

78 312 721

• the smallest number n such that τ (n) ≤ τ (n + 1) ≤ . . . ≤ τ (n + 10): here

8 ≤ 8 ≤ 8 12 ≤ 12 ≤ 12 16 ≤ 16 ≤ 16 ≤ 16 ≤ 16 (see the number 241).

78 436 511

• the

14th

solution of σ2(n) = σ2(n + 2) (see the number 1 079).

79 509 528

• the ninth number n such that β(n)|β(n + 1) and β(n + 1)|β(n + 2): here 54|648

and 648|7950960 (see the number 225 504).

80 314 575

• the smallest number n such that τ (n) τ (n + 1) . . . τ (n + 7): here

48 40 24 20 16 12 8 4 (see the number 45).