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.