A227581 Array r(m,n) = least k such that H(m) + H(n) - H(m*n + k) < g, where H denotes harmonic number and g denotes the Euler-Mascheroni constant.

1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 7, 7, 7, 7, 7, 7, 7

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

Cf. A227582, A227586.

Sequence in context: A023968 A284248 A204166 * A263846 A350088 A178786

Adjacent sequences: A227578 A227579 A227580 * A227582 A227583 A227584

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Jul 17 2013

Status

approved