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).
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