226 Jean-Marie De Koninck

74 848

• the smallest number n such that

f(n + 1) = f(n + 2) = f(n + 3) = f(n + 4) = f(n + 5),

where f(n) stands for the product of the exponents in the factorization of n

(see the number 843).

75 239

• the second prime number q such that

∑

p≤q

p is a multiple of 1 000: the se-

quence of numbers satisfying this property begins as follows: 35 677, 75 239,

81 761, 85 199, 85 531, 86 813, 95 717, . . .

75 329

• the smallest prime number p such that Ω(p − 1) = Ω(p + 1) = 8: here 75 328 =

26

· 11 · 107 and 75 330 = 2 ·

35

· 5 · 31 (see the number 271).

75 361 (= 11 · 13 · 17 · 31)

• the third pseudoprime in bases 2, 3, 5 and 7 (see the number 29 341).

76 544

• the smallest number n such that 9! divides 1+2+ . . . + n (see the number 224);

• the smallest number n such that n and n + 1 are both divisible by a seventh

power: 76 544 = 28 · 13 · 23 and 76 545 = 37 · 5 · 7 (see the number 1 215).

76 571

• the smallest composite number n such that σ(n+56) = σ(n)+56; the composite

numbers 472 601, 929 964, 1 644 236, 3 143 591, 21 887 471 and 28 724 844 are also

solutions of this equation.

77 140

• the smallest 5-composite number n such that n+2 is also a 5-composite number:

here 77 140 =

22

· 5 · 7 · 19 · 29 and 77 142 = 2 · 3 · 13 · 23 · 43 (see the number

1 428).

77 141

• the smallest prime number p such that p − 1 and p + 1 each have exactly five

distinct prime factors: here 77 140 =

22

·5·7·19·29 and 77 142 = 2·3·13·23·43

(see the number 131).