A321047 - OEIS

A321047

a(n) = Sum_{1<=i<j<=n} floor(n*(1/i - 1/j)).

0, 1, 3, 8, 14, 26, 38, 57, 77, 103, 130, 170, 206, 251, 300, 355, 412, 481, 548, 628, 709, 794, 886, 993, 1097, 1207, 1328, 1451, 1578, 1723, 1863, 2018, 2170, 2332, 2500, 2686, 2872, 3063, 3256, 3466, 3672, 3902, 4125, 4358, 4606, 4849, 5103, 5372, 5643, 5920, 6208, 6498, 6802, 7110

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OFFSET

1,3

COMMENTS

In an n-minute race between n competitors numbered 1 to n, where competitor number k runs at speed 1/k rpm for all k, a(n) is the number of overtakings.

LINKS

Table of n, a(n) for n=1..54.

FORMULA

a(n) = Sum_{1<=i<j<=n} floor(n*(1/i-1/j)).

a(n) = Sum_{k=0..n-1} T(n,k)*k, where T is A321368 in bivariate form.

MATHEMATICA

a[n_] := Sum[ Sum[ Floor[n*(1/i - 1/j)], {i, 1 , j} ], {j, 1, n} ]; Array[a, 50] (* Amiram Eldar, Nov 08 2018 *)

PROG

(PARI)

a(n)=sum(y=2, n, sum(x=1, y-1, floor(n*(1/x-1/y))))

for(n=1, 80, print1(a(n), ", "))

CROSSREFS