OFFSET
1,2
COMMENTS
Since log(m*n) = log m + log n and log n is "close to" H(n) - g, this array indicates the "closeness" of H(m*n) to H(m) + H(n). Conjectures:
(1) r(n,n) = n for n >= 1;
(2) 2*H(n) - H(n^2 + n) < g < 2*H(n) - H(n^2 + n - 1);
(3) floor(1/(g - 2*H(n) + H(n^2 + n)) = 6*n*(n+1);
(4) floor(1/(2*H(n) + H(n^2 + n - 1) - g) = A227582(n).
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..1830
EXAMPLE
Northwest corner:
1 2 2 3 3 4 4 5 6 6
2 2 3 3 4 4 5 5 6 6
2 3 3 4 4 5 5 6 6 7
3 3 4 4 5 5 6 6 7 7
3 4 4 5 5 6 6 7 7 8
4 4 5 5 6 6 7 7 8 8
4 5 5 6 6 7 7 8 8 9
r(2,3) = 3 because h(2) + h(3) - h(9) = 0.504... < g = 0.577... < h(2) + h(3) - h(8) = 0.615... .
MATHEMATICA
h[n_] := h[n] = HarmonicNumber[n]; z = 20; r[m_, n_] := Module[{Nn = N[n, 50], Nm = N[m, 50]}, NestWhile[# + 1 &, Floor[(m + n)/2], ! h[Nm] + h[Nn] - h[Nm*Nn + #] < EulerGamma &]]; Table[r[m, n], {m, z}, {n, z}] // TableForm (* array *)
Flatten[Table[r[n - k + 1, k], {n, z}, {k, n, 1, -1}]] (* sequence *)
(* Peter J. C. Moses, Jul 16 2013 *)
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Jul 17 2013
STATUS
approved