Those Fascinating Numbers 257

371 549

• the smallest number which can be written in two distinct ways as the sum of two

co-prime numbers with an

index180

of composition ≥ 5: 371 549 =

28

+

135

=

310

+

22

·

57,

each of these last four numbers having as index of composition

8, 5, 10 and 5.49485 respectively; the sequence of numbers satisfying this pro-

perty begins as follows: 371 549, 1 016 807, 2 657 333, 19 592 147, 143 123 843,

257 400 763, 1 586 054 707, 2 461 855 097, 2 579 381 441, 4 110 348 907, 4 755 143 089,

6 866 235 461, 8 382 781 919, 9 803 235 961, . . . (see the number 34 525 900 789 931).

373 457

• the fourth composite number n such that

2n−2

≡ 1 (mod n); see the number

20 737.

373 649

• the first term of the smallest sequence of ten consecutive prime numbers all of

the form 4n + 1 (see the number 2 593).

378 661

• the largest solution x of the diophantine equation x2 − 17 = y3, namely the one

given by (x, y) = (378661, 5234) (see the number 5 234).

383 846

• the ninth number n such that 2n ≡ −2 (mod n) (see the number 946).

388 961

• the largest known prime number p such that P

(p2

−1) ≤ 17 and P

(p2

−1) = 17;

here

p2

− 1 =

26

·

34

· 5 ·

74

· 11 · 13 · 17 (see the number 4 801).

389 052

• the smallest number which can be written as the sum of the cubes of three

distinct prime numbers and as the sum of the cubes of four distinct prime

numbers: here 389 052 =

23

+

33

+

733

=

73

+

433

+

473

+

593;

the sequence of

numbers satisfying this property begins as follows: 389 052, 493 191, 1 442 932,

19 927 270, 31 867 188, 36 265 069, 49 431 241, 95 444 028, 110 621 861, . . .

180The interest for this number comes from the fact that no one knows if there exists a number

which can be written as the sum of two fifth powers in two distinct ways. This question is however

settled for the fourth powers (see the footnote attached to the number 635 318 657).