A096005 For k >= 1, let b(k) = ceiling( Sum_{i=1..k} 1/i ); a(n) = number of b(k) that are equal to n.

0, 1, 2, 7, 20, 52, 144, 389, 1058, 2876, 7817, 21250, 57763, 157017, 426817, 1160207, 3153770, 8572836, 23303385, 63345169, 172190019, 468061001, 1272321714, 3458528995, 9401256521, 25555264765, 69466411833, 188829284972

Offset

0,3

Links

Table of n, a(n) for n=0..27.

Formula

a(n+1)/a(n) approaches e = exp(1) = 2.71828...

First differences of A002387. - Vladeta Jovovic, Jul 30 2004

Example

The ceilings of the first several partial sums of the reciprocal of the positive integers are 1 2 2 3 3 3 3 3 3 3 4 4 and the series is monotonically increasing, so a(0) = 0 (there being no zero), a(1) = 1 (there being but one 1) and a(3) = 7 (there being seven 3s).

Mathematica

fh[0] = 0; fh[1] = 1; fh[k_] := Module[{tmp}, If[ Floor[tmp = Log[k + 1/2] + EulerGamma] == Floor[tmp + 1/(24k^2)], Floor[tmp], UNKNOWN]]; a[0] = 1; a[1] = 2; a[n_] := Module[{val}, val = Round[Exp[n - EulerGamma]]; If[fh[val] == n && fh[val - 1] == n - 1, val, UNKNOWN]]; Table[ a[n + 1] - a[n], {n, 0, 27}] (* Robert G. Wilson v, Aug 05 2004 *)

Prog

(C)
int A096005(int k)
{    if(k<3) return k;
    double sum = 0, n = 1; int ceiling = 2, cnt = 0;
    for(;; ) {
        sum += 1/n++;
        if(sum < ceiling) { cnt++; continue; }
        if(ceiling++ == k) return cnt; else cnt = 1; }
} /* Oskar Wieland, May 01 2014 */

Crossrefs

Cf. A002387, A096143.

Sequence in context: A134311 A362973 A066373 * A050532 A050513 A348571

Adjacent sequences: A096002 A096003 A096004 * A096006 A096007 A096008

Keyword

nonn

Author

W. Neville Holmes, Jul 29 2004

Extensions

More terms from Robert G. Wilson v, Aug 05 2004

Status

approved