208 Jean-Marie De Koninck

29 341

• the smallest pseudoprime in bases 2, 3, 5 and 7; it is also the smallest pseudo-

prime in bases 2, 3, 5, 7 and 11; the sequence of pseudoprimes in bases 2, 3, 5

and 7 begins as follows: 29341, 46657, 75361, 115921, 162401, . . .

29 355

• the eighth number such that 2n + n2 is prime (see the number 2 007).

29 395

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

have the same prime factors, namely here 2, 3, 5 and 7 (see the numbers 3 777

and 20 154).

29 888

• the smallest number n such that n and n + 1 each have seven prime factors

counting their multiplicity: 29 888 =

26

· 467 and 29 889 =

36

· 41 (see the

number 135).

30 031

• the smallest number of the form p1p2 . . . pk +1 which is not prime: here 30 031 =

2 · 3 · 5 · 7 · 11 · 13 + 1 = 59 · 509.

30 240

• the smallest 4-perfect number, that is a number n such that σ(n) = 4n; the

list of 4-perfect numbers begins as follows: 30 240, 32 760, 2 178 540, 23 569 920,

45 532 800, 142 990 848, 1 379 454 720, 43 861 478 400, 66 433 720 320,

153 003 540 480, 403 031 236 608, . . . ; there seems to exist only 36 4-perfect num-

bers (see R.K. Guy [101], B2).

30 375

• the smallest number n which allows the sum

m≤n

1

φ(m)

to exceed 20 (see the

number 177).