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).