Those Fascinating Numbers 41

128

• the largest number which is not the sum of distinct squares (see R.K. Guy [101],

C20);

• the fourth number n 1 such that φ(σ(n)) = n: the only solutions n

109

of this equation are 1, 2, 8, 12, 128, 240, 720, 6 912, 32 768, 142 560, 712 800,

1 140 480, 1 190 410, 3 345 408, 3 571 200, 5 702 400, 14 859 936, 29 719 872,

50 319 360 and 118 879 488 (see R.K. Guy [101], B42)51.

129

• the smallest number n which allows the sum

m≤n

1

σ(m)

to exceed 4; if we let nk

stand for the smallest number n such that

m≤n

1

σ(m)

k, then n2 = 7, n3 = 29,

n4 = 129, n5 = 5 71, n6 = 2 525, n7 = 11 167, n8 = 49 372, n9 = 218 295 and

n10 = 965 177.

130

• the fifth dihedral perfect number: a number n is said to be dihedral perfect if

τ (n) + σ(n) = 2n: the sequence of numbers satisfying this property begins as

follows: 1, 3, 14, 52, 130, 184, 656, 8 648, 12 008, 34 688, 2 118 656, 33 721 216,

40 575 616, 59 376 256, 89 397 016, 99 523 452, 134 438 912, 150 441 856,

173 706 136, 283 417 216, 537 346 048, 1 082 640 256, . . . : the quantity τ (n) +

σ(n) represents the total number of subgroups of the group of symmetries of a

regular polygon with n sides (see S. Cavior [32]); it is not known if this sequence

of numbers is infinite;

• the second solution of σ2(n) = σ2(n + 11) (see the number 66).

131

• the smallest prime number p such that p − 1 and p + 1 each have exactly three

distinct prime factors: here 130 = 2·5·13 and 132 = 22 ·3·11; if we let p = p(k)

stand for52 the smallest prime number p such that ω(p − 1) = ω(p + 1) = k,

then p(2) = 11, p(3) = 131, p(4) = 1 429, p(5) = 77 141, p(6) = 3 847 271 and

p(7) = 117 048 931.

132

• the sixth Catalan number (see the number 14).

51It

is clear that any solution n 1 of φ(σ(n)) = n is even, since φ(n) is even for all n ≥ 3.

52It would be interesting if one could prove that this sequence is infinite.