Those Fascinating Numbers 71

309

• the number of digits in the decimal expansion of the Fermat number

2210

+ 1.

312

• the smallest number n such that φ(n) + σ(n) = 3n: the only numbers n

1010 satisfying this property are 312, 560, 588, 1 400, 85 632, 147 492, 556 160,

569 328, 1 590 816, 2 013 216, 3 343 776, 4 695 456, 9 745 728, 12 558 912,

22 013 952, 23 336 172, 30 002 960, 52 021 242, 75 007 400, 137 617 728,

153 587 720, 699 117 024, 904 683 264, 2 468 053 248 and 2 834 395 104 (see R.K.

Guy [102]; see also the number 23 760).

313

• the largest prime factor of the largest known unitary perfect number (see the

number 6);

• the second prime number of the form (x4 + y4)/2: here 313 = (54 + 14)/2 (see

the number 41);

• the smallest number n such that φ(n) φ(n +1) φ(n +2): here 312 156

144 (see the numbers 105 and 823).

314

• the number of digits in the decimal expansion of the 13th perfect number

2520(2521 − 1).

317

• one of the nine known numbers k such that 11 . . . 1

k

is prime (H.C. Williams

[203]); see the number 19;

• the seventh prime number pk such that p1p2 . . . pk − 1 is prime: the only known

prime numbers satisfying this property are 3, 5, 11, 13, 41, 89, 317, 337, 991,

1 873, 2 053, 2 377, 4 093, 4 297, 4 583, 6 569, 13 033 and 15 877.

318

• the smallest number n for which the sequence n, t(n), t(t(n)), . . . is not

bounded: here

t(n) = n

p|n

1 +

1

p

− n

represents the sum of the divisors d of n such that 1 ≤ d n and such that

n/d is square-free (Problem #10323, Amer. Math. Monthly 103, no.8, Oct.

1996, 697-698).