Those Fascinating Numbers 63

m, m + 1, m + 2, . . . , m + k − 1 are all divisible by an th power, here are the

values of some of the mk,:

(a star next to a number indicates that it is “most likely” the smallest with

that property)

= 2 = 3 = 4

k = 2 8 80 80

k = 3 48 1 375 33 614

k = 4 242 22 624 202 099 373

k = 5 844 18 035 622 40 280 549 372

k = 6 22 020 4 379 776 620 430 995 495 889 374(*)

k = 7 217 070 1 204 244 328 624(*) 77 405 340 617 896 874(*)

= 5 = 6

k = 2 1 215 16 767

k = 3 2 590 623 26 890 623

k = 4 2 146 909 373 1 507 545 109 375

k = 5 105 636 978 090 621 777 562 026 420 218 750(*)

k = 6 3 269 698 976 575 137 500(*) 283 435 321 166 212 288 109 372(*)

= 7

k = 2 76 544

k = 3 2 372 890 624

k = 4 390 491 792 890 623(*)

k = 5 2 083 234 733 888 734 218 749(*)

k = 6 18 962 123 650 219 836 035 505 781 245(*)

• the smallest number n having the same number of divisors as the three numbers

that follow it: τ (242) = τ (243) = τ (244) = τ (245) = 6; the sequence of num-

bers satisfying this property begins as follows: 242, 3655, 4503, 5943, 6853,

7256, 8392, 9367, 10983, 11605, 11606, . . . (see the number 33);

• the smallest number n such that the decimal expansion of

2n

contains three

consecutive zeros (see the number 53).

243

• the seventh number n such that

n|2n

+ 1; the sequence of numbers satisfying

this property begins as follows: 1, 3, 9, 27, 81, 171, 243, 513, 729, 1539, 2187,

3249, 4617, 6561, 9747, 13203, 13851, 19683, 29241, 39609, 41553,. . .

245

• the number of pseudoprimes in base 2 which do not exceed

106;

if we denote

by N(x) the number of pseudoprimes in base 2 which do not exceed x, then we

have the following table:

k 1 2 3 4 5 6 7 8 9

N(10k) 0 0 3 22 78 245 750 2 057 5 597