Those Fascinating Numbers 123
1 260
the smallest number n such that φ(n + 1) = 4φ(n); the sequence of numbers
satisfying this property begins as follows: 1260, 13650, 17556, 18720, 24510,
42120, 113610, 244530, 266070, 712080, . . . (see also the number 11 242 770); if
nk stands for the smallest number n such that φ(n + 1) = kφ(n), then n1 = 1,
n2 = 2, n3 = 6, n4 = 1 260 and n5 = 11 242 770 (for the analogue problem with
equation φ(n) = kφ(n + 1), see the number 629);
the smallest vampire number: a number n with 2r digits, r 2, is called a
vampire number if it can be written as the product of two numbers a and b (each
having r digits) and such that the set of numbers formed by joining the digits
of a and b is the same set as that of the digits of n (in this case, 1260 = 21 · 60);
there exist seven four digit vampire numbers, namely 1260, 1395, 1435, 1530,
1827, 2187 and 6880, while there are 155 six digit vampire numbers and 3382
eight digit vampire numbers (see E.W. Weisstein [201], p. 1894);
the
16th
highly composite number (see the number 180).
1 263
the largest number of the form 8k + 7 which can be written as the sum of
exactly three powerful numbers in exactly two distinct ways: here 1 263 =
72 + 53 + 33 · 112 = 22 · 72 + 23 · 72 + 33 · 52 (see the number 118).
1 270
the rank of the prime number which appears the most often as the
16th
prime
factor of an integer: p1270 = 10 343 (see the number 199).
1 279
the exponent of the
15th
Mersenne prime
21 279
1 (Robinson, 1952).
1 290
the second solution of σ2(n) = σ2(n + 12) (see the number 864).
1 291
the largest solution n of
n3
n +3 =
3m2:
the only solutions (n, m) of
n3
n +
3 =
3m2
are (n, m) = (−1, 1), (1,1), (3,3), (11,21), (13,27) and
(171,1291).124
124The interest for equation n3 n + 3 = 3m2 comes from the quest for the solutions (r, s) of
equation 1 + 2 + . . . + r =
12
+
22
+ . . . +
s2,
that is of equation
3(r2
+ r) = s(s + 1)(2s + 1), which
becomes n3 n + 3 = 3m2 after setting m = 2r + 1 and n = 2s + 1. R. Finkelstein & H. London
[81] proved that the only integer solutions of equation
n3
n + 3 =
3m2
are those mentioned above,
which yields the solutions (r, s) = (1, 1), (10,5), (13,6) and (645,85) of 3(r2 + r) = s(s + 1)(2s + 1).
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