Those Fascinating Numbers 77
348
the smallest number n such that n, n +2, n +4, n +6 are abundant (we say that
a number n is abundant if σ(n) 2n): the sequence (nk)k≥1 of the smallest
numbers n = nk such that n, n + 2, . . . , n + 2k are abundant begins as follows:
18, 100, 348, 2 988, 801 340, . . . ;
the smallest solution of τ (n + 12) = τ (n) + 12; the sequence of numbers satis-
fying this equation begins as follows: 348, 492, 758, 788, 852, 898, 933, 1164,
1212, 1236, . . .
353
the smallest number whose fourth power can be written as the sum of four
fourth powers: 3534 = 304 + 1204 + 2724 + 3154 (R.Norrie, 1911): see R.K. Guy
[101], D1;
the third prime number of the form
(x4
+
y4)/2:
here 353 =
(54
+
34)/2
(see
the number 41).
357
the second square-free composite number n for which p|n =⇒ p + 3|n + 3 (see
the number 165).
359
the smallest prime factor of the Mersenne number
2179
1, whose complete
factorization is given by
2179
1 = 359 · 1433 · 1489459109360039866456940197095433721664951999121.
360 (=
23
·
32
· 5)
the smallest number m such that equation σ(x) = m has exactly nine solutions,
namely 120, 174, 184, 190, 267, 295, 319, 323 and 359;
the smallest solution of
σ(n)
n
=
13
4
; the sequence of numbers satisfying this
equation begins as follows: 360, 2 016, 1 571 328, 4 428 914 688,. . . ;
the 13th highly composite number (see the number 180).
361
the smallest Smith number (see the number 22) which is a perfect square:
361 = 192 and 3 + 6 + 1 = 10 = 1 + 9;
the number at which the function (ψ(x) θ(x))/

x reaches its maximal value,
where θ(x) =
p≤x
log p and ψ(x) =
pm
≤x
m≥1
log p (J.B. Rosser & L. Schoenfeld [178]).
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