Those Fascinating Numbers 65
258
the rank of the prime number which appears the most often as the
12th
prime
factor of an integer : p258 = 1 627 (see the number 199).
261
the only three digit number n such that
2n
n is prime (an observation due
to Meng Hsuan Wu (2002)): the only known numbers n for which the corre-
sponding number
2n
n is prime are 13, 19, 21, 55, 261, 3 415, 4 185, 7 353 and
12 213.
263
the second prime number p such that
79p−1
1 (mod
p2):
the only prime
numbers p
232
satisfying this congruence are 7, 263, 3 037, 1 012 573 and
60 312 841 (see Ribenboim [169], p. 347);
the smallest prime factor of the Mersenne number
2131
1, whose complete
factorization is given by
2131
1 = 263 · 10350794431055162386718619237468234569.
264
the smallest number which can be written in two distinct ways as the sum of
positive powers of its digits: 264 = 21 + 61 + 44 = 25 + 63 + 42: the only other
number satisfying this property is 373;
the largest three digit number = 100, 200, 300, and whose square contains no
more than two distinct digits: 2642 = 69 696 (see the number 109);
the second number which is not a palindrome, but whose square is a palindrome
(see the number 26).
265
the number of possible arrangements of the integers 1, 2, 3, 4, 5, 6 with the
restriction that the integer j must not be in the
jth
position for each j, 1
j 6: more generally, the number nk of possible arrangements of the integers
1, 2, . . . , k with the restriction that the integer j must not be in the
jth
position
for each j, 1 j k, is equal to
k!
1
2!

1
3!
+ . . . +
(−1)k
k!
,
and therefore n2 = 1, n3 = 2, n4 = 9, n5 = 44, n6 = 265, n7 = 1 854,
n8 = 14 833, n9 = 133 496, n10 = 1 334 961 and n11 = 14 684 570;
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