270 Jean-Marie De Koninck

766 261

• the first term of the smallest sequence of 11 consecutive prime numbers all of

the form 4n + 1 (see the number 2 593).

768 594

• the fifth number n such that σ(n), σ(n + 1), σ(n + 2), σ(n + 3) and σ(n + 4)

all have the same prime factors, namely here the primes 2, 3, 5, 7 and 61 (see

the numbers 3 777 and 20 154).

783 700

• the third multiple of 100 for which the next 100 numbers include exactly

17 prime numbers, namely 783701, 783703, 783707, 783719, 783721, 783733,

783737, 783743, 783749, 783763, 783767, 783779, 783781, 783787, 783791,

783793 and 783799 (see the number 400).

798 644

• the tenth number which is not a palindrome, but whose square is a palindrome:

here 798 6442 = 637832238736 (see the number 26).

801 340

• the smallest abundant number n such that n + 2, n + 4, n + 6, n + 8 and n + 10

are also abundant (see the number 348).

806 095

• the number of Niven numbers ≤

107

(see the number 213).

811 538

• the smallest number which can be written as the sum of three distinct fourth

powers in three distinct ways:

811 538 =

44

+

234

+

274

=

74

+

214

+

284

=

124

+

174

+

294,

a representation first obtained in 1966 by Lander & Parkin [122] (see the num-

ber 6 578).

815 654

• the

12th

number n such that φ(n)σ(n) is a fourth power: here

φ(815654)σ(815654) =

8404

(see the number 170).