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
Keywords
Examples
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).
Programs
-
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 */
-
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 *)
Formula
a(n+1)/a(n) approaches e = exp(1) = 2.71828...
First differences of A002387. - Vladeta Jovovic, Jul 30 2004
Extensions
More terms from Robert G. Wilson v, Aug 05 2004