Those Fascinating Numbers 127
1 386
the smallest number n which allows the sum
m≤n
1
φ(m)
to exceed 14 (see the
number 177).
1 393
the smallest number which can be written both as the sum of three fourth
powers and as the sum of four fourth powers: 1393 =
24
+
34
+
64
=
44
+
44
+
44
+
54.
1 395
the second vampire number; in this case, 1 395 = 93·15 (see the number 1 260).
1 399
the smallest prime factor of the Mersenne number
2233
1, whose complete
factorization is given by
2233
1 = 1399 · 135607 · 622577
·116868129879077600270344856324766260085066532853492178431.
1 400
the fourth number n such that φ(n) + σ(n) = 3n (see the number 312);
the second multiple of 100 such that the next 100 numbers contain exactly 17
prime numbers, namely 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453,
1459, 1471, 1481, 1483, 1487, 1489, 1493 and 1499 (see the number 400).
1 415
the fourth composite number n such that σ(n + 6) = σ(n) + 6 (see the number
104).
1 428
the smallest 4-composite number n such that n+2 is also a 4-composite number:
we say that a number n is k-composite if ω(n) = k: here 1428 =
22
· 3 · 7 · 17
and 1430 = 2 · 5 · 11 · 13; denoting by nk the smallest k-composite number n
such that n + 2 is also k-composite, we have the following table:
k 1 2 3 4 5 6 7 8
nk 2 10 130 1 428 77 140 1 456 728 117 048 930 10 326 137 820
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