150 Jean-Marie De Koninck
2 657
the tenth prime number pk such that p1p2 . . . pk + 1 is prime (see the number
379).
2 667
the smallest number larger than 1 and whose sum of divisors is a sixth power:
σ(2667) = 46.
2 673
the smallest number which can be written as the sum of three fourth powers in
two distinct ways: 2 673 =
24
+
44
+
74
=
34
+
64
+
64
(see the number 6 578).
2 685
the seventh solution of σ(n) = σ(n + 1) (see the number 206).
2 687
the smallest prime factor of the Mersenne number
279
1, whose complete
factorization is given by
279
1 = 2687 · 202029703 · 1113491139767.
2 693
the smallest prime number p such that 12p−1 1 (mod p2): the only other
prime number p 232 satisfying this congruence is 123 653 (see Ribenboim
[169], p. 347).
2 699
the rank of the prime number which appears the most often as the
18th
prime
factor of an integer: p2699 = 24 251 (see the number 199).
2 719
the largest odd number n 150 000 which cannot be written in the form
x2
+
y2
+
10z2
(see H.G. Gupta [100]).
2 728 (=
23
· 11 · 31)
the second number n such that β(n)|n, β(n + 1)|(n + 1), β(n + 2)|(n + 2) and
β(n + 3)|(n + 3): indeed, here β(2728) = 44, β(2729) = 2729, β(2730) = 30
and β(2731) = 2731; the sequence of numbers satisfying this property begins
as follows: 29, 2728, 3526, 103966, 150587, 4743197, . . .
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