A227581 - OEIS

%I #7 Feb 22 2014 09:23:53

%S 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,

%T 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,

%U 7,7,7,7,7,7,7,7,7,7,8,7,7,7,7,7,7,7

%N 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.

%C 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:

%C (1)  r(n,n) = n for n >= 1;

%C (2)  2*H(n) - H(n^2 + n) < g < 2*H(n) - H(n^2 + n - 1);

%C (3)  floor(1/(g - 2*H(n) + H(n^2 + n)) = 6*n*(n+1);

%C (4)  floor(1/(2*H(n) + H(n^2 + n - 1) - g) = A227582(n).

%H Clark Kimberling, <a href="/A227581/b227581.txt">Table of n, a(n) for n = 1..1830</a>

%e Northwest corner:

%e 1 2 2 3 3 4 4 5 6 6

%e 2 2 3 3 4 4 5 5 6 6

%e 2 3 3 4 4 5 5 6 6 7

%e 3 3 4 4 5 5 6 6 7 7

%e 3 4 4 5 5 6 6 7 7 8

%e 4 4 5 5 6 6 7 7 8 8

%e 4 5 5 6 6 7 7 8 8 9

%e r(2,3) = 3 because h(2) + h(3) - h(9) = 0.504... < g = 0.577... < h(2) + h(3) - h(8) = 0.615... .

%t 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 *)

%t Flatten[Table[r[n - k + 1, k], {n, z}, {k, n, 1, -1}]]  (* sequence *)

%t (* _Peter J. C. Moses_, Jul 16 2013 *)

%Y Cf. A227582, A227586.

%K nonn,tabl,easy

%O 1,2

%A _Clark Kimberling_, Jul 17 2013