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).
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