Those Fascinating Numbers 17

46

• the rank of the prime number which appears the most often as the eighth prime

factor of an integer: p46 = 199 (see the number 199).

47

• the prime number which appears the most often as the sixth prime factor of an

integer (see the number 199);

• the fifth Hamilton number (see the number 923).

48

• the smallest number which is divisible by a square 1 and is followed by two

other numbers with the same property: here 48 =

24

·3, 49 =

72

and 50 =

2·52;

the sequence of numbers satisfying this property begins as follows: 48, 98, 124,

242, 243, 342, 350, 423, 475, 548, 603, 724, 774, 844, 845, 846,. . . ;

• the smallest number for which the product of its proper divisors is a fourth

power, that is such that

d|n, dn

d =

a4:

here

2 · 3 · 4 · 6 · 8 · 12 · 16 · 24 = 5 308 416 =

484;

• the second number n divisible by a square 1 and such that γ(n+1)−γ(n) = 1;

the sequence of numbers satisfying this property begins as follows: 8, 48, 224,

960, 65 024, 261 120, 1 046 528, 4 190 208, . . .

23

(see the number 98).

49

• the smallest number n divisible by a square 1 and such that δ(n+1)−δ(n) =

1, where δ(n) =

p n

p; the sequence of numbers satisfying this property begins

as follows: 49, 1 681, 18 490, 23 762, 39 325, 57 121, 182 182, 453 962, 656 914,

843 637, . . .

24;

• the second solution w + s of the aligned houses problem (see the number 35).

23It is easy to see that each number n = 2r+1(2r−1 −1), where 2r −1 and 2r−1 −1 are square-free,

is a solution of γ(n + 1) − γ(n) = 1. The numbers r 200 such that

2r

− 1 and

2r−1

− 1 are both

square-free are the numbers 2, 3, 4, 5, 8, 9, 10, 11, 14, 15, 16, 17, 23, 26, 27, 28, 29, 32, 33, 34, 35,

38, 39, 44, 45, 46, 47, 50, 51, 52, 53, 56, 57, 58, 59, 62, 65, 68, 69, 70, 71, 74, 75, 76, 77, 82, 83, 86,

87, 88, 89, 92, 93, 94, 95, 98, 99, 104, 107, 112, 113, 116, 117, 118, 119, 122, 123, 124, 125, 128, 129,

130, 131, 134, 135, 142, 143, 146, 149, 152, 153, 154, 158, 159, 164, 165, 166, 167, 170, 171, 172,

173, 176, 177, 178, 179, 182, 183, 184, 185, 188, 191, 194, 195, 196 and 197. Therefore it follows

that equation γ(n + 1) − γ(n) = 1 has infinitely many solutions n (with µ(n) = 0). But it is not

known if there exist infinitely numbers of the form

2r

− 1 which are square-free. In fact, Andrew

Granville believes (private communication) that it is unlikely that one could easily prove that there

exist infinitely many square-free numbers of the form

2r

− 1, since if that was the case, it would

follow that there exist infinitely many prime numbers which are not Wieferich primes, a result that

is certainly true, but that we are so far unable to prove without assuming the abc Conjecture (see

Silverman [186]).

24There

exist infinitely many numbers n divisible by a square and satisfying δ(n + 1) − δ(n) = 1.

This follows from the fact that the Fermat-Pell equation 2x2 − y2 = 1 has infinitely many solutions.