A252229 The number of numbers j*r^k in the interval [n,n+1), where r = (1 + sqrt(5))/2, the golden ratio, and j >=0, k >= 0.

1, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 3, 2, 2, 3, 2, 3, 2, 4, 2, 2, 2, 4, 3, 3, 2, 2, 3, 2, 2, 3, 3, 2, 3, 2, 4, 3, 2, 2, 3, 2, 2, 3, 3, 4, 2, 2, 3, 3, 2, 3, 2, 3, 2, 2, 3, 3, 3, 2, 2, 3, 3, 2, 2, 3, 4, 3, 2, 2, 3, 2, 3, 2, 3, 2

Offset: 0

Author

Clark Kimberling, Dec 16 2014

Keywords

• nonn
• easy

Comments

The least n for which a(n) = 4 is 29; the least n for which a(n) = 5 is 199.

Examples

			in [0,1):  0
in [1,2):  1, 1 + r
in [2,3):  2, 2 + r
in [3,4):  3, 1+2*r
in [4,5):  4, 1+3*r, 2 + r
		

Links

• Clark Kimberling, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A182801, A020959.

Programs

Mathematica

z = 100; r = (1 + Sqrt[5])/2;
s[n_, j_] := s[n, j] = Floor[Log[n/j]/Log[r]];
a[n_] := a[n] = Sum[s[n + 1, j] - s[n, j], {j, 1, Floor[(n + 1)/r]}];
t = Join[{1}, Table[1 + a[n], {n, 1, z}]] (* A252229 *)

Formula

a(n) = 1 + sum{s(n+1,j) - s(n,j), j=1..floor[(n+1)/r]}, where s(n,j) = floor[log(n/j)/log(r)], for n >= 1.