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A252229
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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.
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2
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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
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OFFSET
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0,2
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COMMENTS
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The least n for which a(n) = 4 is 29; the least n for which a(n) = 5 is 199.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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