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A190504 n+[ns/r]+[nt/r]+[nu/r]; r=golden ratio, s=r+1, t=r+2, u=r+3. 4


%S 6,14,21,29,38,45,52,59,68,76,83,91,100,106,114,121,130,138,145,153,

%T 159,168,176,183,191,200,207,214,221,230,238,245,253,262,268,276,283,

%U 291,300,307,315,321,330,338,345,353,362,368,376,383,392,400,407,415,421,430,438,445,453,462,469,476,483,492,500,507,515,524,530

%N n+[ns/r]+[nt/r]+[nu/r]; r=golden ratio, s=r+1, t=r+2, u=r+3.

%C This is one of four sequences that partition the positive integers. In general, suppose that r, s, t, u are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1, {h/u: h>=1} are pairwise disjoint. Let a(n) be the rank of n/r when all the numbers in the four sets are jointly ranked. Define b(n), c(n), d(n) as the ranks of n/s, n/t, n/u, respectively. It is easy to prove that

%C a(n)=n+[ns/r]+[nt/r]+[nu/r],

%C b(n)=n+[nr/s]+[nt/s]+[nu/s],

%C c(n)=n+[nr/t]+[ns/t]+[nu/t],

%C d(n)=n+[nr/u]+[ns/u]+[nt/u], where []=floor.

%C Taking r=golden ratio, s=r+1, t=r+2, u=r+3 gives

%C a=A190504, b=A190505, c=A190506, d=A190507.

%t r=GoldenRatio; s=r+1; t=r+2; u=r+3;

%t a[n_] := n + Floor[n*s/r] + Floor[n*t/r]+Floor[n*u/r];

%t b[n_] := n + Floor[n*r/s] + Floor[n*t/s]+Floor[n*u/s];

%t c[n_] := n + Floor[n*r/t] + Floor[n*s/t]+Floor[n*u/t];

%t d[n_] := n + Floor[n*r/u] + Floor[n*s/u]+Floor[n*t/u];

%t Table[a[n], {n, 1, 120}] (*A190504*)

%t Table[b[n], {n, 1, 120}] (*A190505*)

%t Table[c[n], {n, 1, 120}] (*A190506*)

%t Table[d[n], {n, 1, 120}] (*A190507*)

%Y Cf. A190505, A190506, A190507 (the other three sequences in the partition of N).

%K nonn

%O 1,1

%A _Clark Kimberling_, May 11 2011

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Last modified September 24 21:51 EDT 2022. Contains 356949 sequences. (Running on oeis4.)