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].
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