Those Fascinating Numbers 49

• perhaps the number n for which eπ

√

n is the closest to an integer: indeed,

eπ

√

163 is remarkably close to an integer, namely to 262537412640768744 since

it is equal to

262537412640768743.99999999999925007 . . .

(see Rouse Ball & Coxeter [13]).

164

• the fifth solution of φ(n) = φ(n + 1) (see the number 15).

165 (= 3 · 5 · 11)

• the smallest square-free composite number n such that p|n =⇒ p + 3|n + 3: the

sequence of numbers satisfying this property begins as follows: 165, 357, 1885,

2397, 3965, 9447, . . . (see the number 399);

• the

100th

square-free number (the 10 smallest being 2, 3, 5, 6, 7, 10, 11, 13, 14

and 15); if we denote by nk the 10k-th square-free number62, then n1 = 15, n2 =

165, n3 = 1 639, n4 = 16 447, n5 = 164 499, n6 = 1 644 919, n7 = 16 449 370,

n8 = 164 493 391 and n9 = 1 644 934 082.

166

• the

12th

number n such that n!−1 is prime: the only known numbers satisfying

this property are 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546,

974, 1 963, 3 507, 3 610, 6 917, 21 480 and 34 790.

167

• the smallest prime factor of the Mersenne number

283

− 1, whose complete

factorization is given by

283

− 1 = 167 · 57 912 614 113 275 649 087 721;

• the seventh number n such that n! + 2n − 1 is prime (see the number 6 247).

62Since

the number of square-free numbers ≤ x is asymptotic to

(6/π2)x

as x → ∞, it is interesting

to observe that the predicted values mk := (π2/6)10k (where y stands for the nearest integer

to y) differ very little from the actual values of nk: indeed, if we set dk := nk − mk , then one will

notice that d1 = −1, d2 = 1, d3 = −6, d4 = −2, d5 = 6, d6 = −15, d7 = 29, d8 = −16 and d9 = 15.