if

1/1 + 1/2 + 1/3 + ... + 1/(m-1) < n < 1 + 1/2 + 1/3 + ... + 1/(m)

for example

A(1) = 1

A(2) = 4

1/1 + 1/2 + 1/3 < 2 < 1/1 + 1/2 + 1/3 + 1/4

A(3) = 11

A(4) = 31

A(5) = 83

A(6) = 227

A(7) = 614

conjecture

lim A(n+1)/A(n) = e

for example

A(2)/A(1) = 4

A(3)/A(2) = 2.75

A(4)/A(3) = 2.82

A(5)/A(4) = 2.68

A(6)/A(5) = 2.73

A(7)/A(6) = 2.70