Those Fascinating Numbers 145
2 311
the fifth prime number p of the form p =
r
i=1
pi + 1: 2 311 = 2 · 3 · 5 · 7 · 11 + 1
(see the number 379).
2 312
the third powerful number which can be written as the sum of two co-prime
3-powerful numbers = 1: 2 312 = 125 + 2 187, that is
23
·
172
=
53
+
37
(see the
number 841).
2 318
the smallest number n which allows the sum
m≤n
1
φ(m)
to exceed 15 (see the
number 177).
2 319
the fourth number n such that
2n
7 is prime (see the number 39).
2 329
the smallest number n such that
φ11(n)
= 2, where
φ11(n)
stands for the
11th
iteration of the φ function (see the number 137).
2 351
the smallest prime factor of the Mersenne number 247 1 (Euler, 1741), whose
complete factorization is given by
247
1 = 140 737 488 355 327 = 2351 · 4513 · 13264529;
(probably) the largest number which cannot be written as the sum of two co-
prime numbers whose index of composition is 1.6 (see the number 933).
2 352
the second solution of σ(n) = 3n + 12 (see the number 780).
2 377
the 12th known prime number pk such that p1p2 . . . pk 1 is prime (see the
number 317).
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