274 Jean-Marie De Koninck
1 006 003
the second and largest prime number p 232 such that 3p−1 1 (mod p2);
the smallest is p = 11 (see Ribenboim [169], p. 347); the above congruence is
sometimes called the Mirimanoff congruence.
1 012 321
the smallest seven digit prime number whose digits are consecutive (see the
number 67).
1 013 724
the smallest number n such that ω(n)+ω(n+1)+ω(n+2)+ω(n+3) = 18: here
1 013 724 = 22 ·32 ·29·971, 1 013 725 = 52 ·23·41·43, 1 013 726 = 2·7·19·37·103
and 1 013 727 = 3 · 11 · 13 · 17 · 139 (see the number 987).
1 016 807
the second number which can be written in two distinct ways as the sum of
two co-prime numbers each with an index of composition 5: 1 016 807 =
217
+
311
· 5 =
75
+
26
·
56,
each of these last four numbers having as index of
composition 17, 5.05684, 5 and 6 respectively (see the number 371 549).
1 023 467
the smallest prime number made up of seven distinct digits (see the number
1 039).
1 023 587 (= 17 · 19 · 3169)
the number n which allows the sum
m≤n
Ω(m)=3
1
m
to exceed 3 (see the number 402).
1 025 273
the third prime number p such that
41p−1
1 (mod
p2)
(see the number 29).
1 032 256
the only solution n
1012
of σ(n) = 2n + 127.
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