Those Fascinating Numbers 131

1 493

• the sixth Stern number (see the number 137).

1 518

• the smallest number n such that P (n + i) ≤

√

n + i, i = 0, 1, 2, 3; if nk stands

for the smallest number n such that P (n + i) ≤

√

n + i, i = 0, 1, 2, . . . , k − 1,

then n4 = 1 518, n5 = 5 828, n6 = 28 032, n7 = n8 = 290 783, n9 = 1 258 500

and n10 = 4 325 170; on the other hand, if mk stands for the smallest number

m such that P (m + i) ≤ (m +

i)1/k,

for i = 0, 1, 2, then m2 = 48, m3 = 134 848

and m4 = 116 026 273.

1 520

• the only number n such that the set A := {1, 15, 24, n} is a diophantine quadru-

plet, that is such that xy + 1 is a perfect square for all x, y ∈ A, x = y (see the

number 120 and its footnote, as well as the number 528).

1 521 (= 32 · 132)

• the smallest number n divisible by a perfect square 1 such that γ(n + 18) =

γ(n) + 18 (and there are no others 108): here we have γ(n + 18) − γ(n) =

3 · 19 − 3 · 13 = 18;

• the third odd number n 1 such that γ(n)|σ(n) (see the number 135).

1 536

• the tenth Granville number (see the number 126).

1 537

• the tenth Keith number (see the number 197).

1 538

• the number of Niven numbers ≤ 10 000 (see the number 213).

1 540

• one of the five numbers (the others are 1, 10, 120 and 7 140) which are both

triangular and tetrahedral (see the number 10).