Those Fascinating Numbers 43

136

• the fifth number n such that φ(n − 1)σ(n − 1) = φ(n)σ(n); the sequence

of numbers satisfying this equation begins as follows: 6, 56, 57, 124, 136,

148, 176, 305, 352, 645, 1 016, 2 465, 19 305, 61 132, 162 525, 476 672, 567 645,

712 725, 801 945, 2 435 489, 3 346 400, 3 885 057, 4 556 000, 8 085 561, 8 369 361,

12 516 693, 22 702 120, 29 628 801, . . . (see the number 55).

137

• the smallest possible value of the largest prime factor of n4 + 1 for n ≥ 4: this

lower bound is reached when n = 10 (see M. Mabkhout [130] as well as the

number 239);

• the second Stern number: a number n is called a Stern number if it cannot

be written as n = p +

2a2

for some prime number p and some number a: it

seems that there exist only eight Stern numbers, namely 17, 137, 227, 977,

1 187, 1 493, 5 777 and 5 993 (only these last two are not prime numbers): see

L. Hodges [111];

• one of the only two prime numbers p (the other being 73) with the

property54

that any number of the form abcdabcd is divisible by

p;55

• the smallest number n such that

φ7(n)

= 2, where

φ7(n)

stands for the seventh

iteration of the φ function; if we consider the sequence (nk)k≥1 defined by

nk = min{n :

φk(n)

= 2}, the first terms of that sequence are 3, 5, 11, 17,

41, 83, 137, 257, 641, 1097, 2329, 4369, 10537, . . . (see Giblin [89], p. 117 and

R.K. Guy [101], B41);

• the fifth prime number p such that

19p−1

≡ 1 (mod

p2)

(see the number 43);

• the largest prime factor of 123456787654321.

139

• the smallest prime number p such that p + 10 is prime and such that each

number between p and p + 10 is composite ; if qk = pr stands for the smallest

prime number such that pr+1 − pr = 10k, we have the following table:

54This simply follows from the fact that 137 · 73 = 10 001.

55Other

interesting properties of the number 137 are mentioned in the book of Martin Gardner

[88].