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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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 A178786 A000196

Adjacent sequences:  A227578 A227579 A227580 * A227582 A227583 A227584

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Jul 17 2013

STATUS

approved

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Last modified December 13 09:05 EST 2017. Contains 295957 sequences.