118 Jean-Marie De Koninck

1 039

• the smallest prime number made up of four distinct digits; if qk stands for the

smallest prime number made up of k distinct digits, then q1 = 2, q2 = 13,

q3 = 103, q4 = 1 039, q5 = 10 243, q6 = 102 359, q7 = 1 023 467, q8 = 10 234 589

and q9 = 102 345 689; it is clear that q10 does not exist (see the number 9 871

for the analogue question for the largest k digit prime number).

1 061

• the number of four digit prime numbers (see the number 21);

• the smallest irregular prime larger than 1000 (see the number 59).

1 069

• the smallest number n requiring eight iterations of the σI (n) function in order

to reach 1: indeed, σI (1069) = 1070, σI (1070) = 648, σI (648) = 121, σI (121) =

133, σI (133) = 160, σI (160) = 6, σI (6) = 4 and σI (4) = 1 (see the number

193).

1 072

• the smallest number which can be written respectively as the sum of two, three

and four distinct cubes: 1 072 = 73 + 93 = 23 + 43 + 103 = 13 + 63 + 73 + 83;

the sequence of numbers satisfying this property begins as follows: 1 072, 6 867,

6 984, 8 576, 9 288, 9 728, 10 261, 10 656, 10 745, 10 773, 10 989, . . . (see the

number 4 802).

1 079

• the fourth solution of σ2(n) = σ2(n+2): the three smallest solutions are 24, 215

and 280; it is mentioned in the book of R.K. Guy [101], B13, that, according to

P. Erd˝ os, the above equation has only a finite number of solutions; nevertheless

one can prove (see J.M. De Koninck [45]) that if Hypothesis H is true (see its

statement on page xvii), then the above equation has infinitely many solutions;

there are 24 solutions

109,

namely 24, 215, 280, 1 079, 947 519, 1 362 239,

2 230 271, 14 939 999, 19 720 007, 32 509 439, 45 581 759, 45 841 247, 49 436 927,

78 436 511, 82 842 911, 101 014 631, 166 828 031, 225 622 151, 225 757 799,

250 999 559, 377 129 087, 554 998 751, 619 606 439 and 846 765 431;

• the smallest number r which has the property that each number can be written

as x1

10

+ x2

10

+ . . . + xr

10,

where the xi’s are non negative integers (see the

number 4).