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.
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