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A184920 a(n) = n+[s*n/r]+[t*n/r]+[u*n/r], where []=floor and r=2^(1/2), s=r+1, t=r+2, u=r+3.

%I A184920 #10 Jun 12 2025 08:35:43
%S A184920 7,15,24,31,40,48,55,64,73,82,89,97,106,113,122,130,140,147,155,164,
%T A184920 171,180,188,195,205,213,222,229,238,246,253,262,271,280,287,295,304,
%U A184920 311,320,328,335,345,353,362,369,378,386,393,402,411,420,427,435,444,451,460,468,478,485,493,502,509,518,526,533,543,551,560,567,575,584,591,600,608,618,625,633,642,649,658,666,673,683,691,700,707,716,724,731,740,749,758,765,773,782,789,798,806,816,823,831,840,847,856,864,871,880,889,898,905,913,922,929,938,946,956,963,971,980,987
%N A184920 a(n) = n+[s*n/r]+[t*n/r]+[u*n/r], where []=floor and r=2^(1/2), s=r+1, t=r+2, u=r+3.
%C A184920 The sequences A184920-A184923 partition the positive integers:
%C A184920   A184920: 7,15,24,31,40,48,55,64,...
%C A184920   A184921: 3,8,13,18,23,27,32,37,...
%C A184920   A184922: 2,5,9,12,16,19,22,26,29,...
%C A184920   A184923: 1,4,6,10,11,14,17,20,21,...
%C A184920 Jointly rank the sets {h*r}, {i*s}, {j*t}, {k*u}, where h>=1, i>=1, j>=1, k>=1.
%C A184920 The position of n*r in the joint ranking is n+[s*n/r]+[t*n/r]+[u*n/r], and likewise for the positions of n*s, n*t, and n*u.
%t A184920 r=2^(1/2); s=r+1; t=r+2; u=r+3;
%t A184920 a[n_]:=n+Floor[n*s/r]+Floor[n*t/r]+Floor[n*u/r];
%t A184920 b[n_]:=n+Floor[n*r/s]+Floor[n*t/s]+Floor[n*u/s];
%t A184920 c[n_]:=n+Floor[n*r/t]+Floor[n*s/t]+Floor[n*u/t];
%t A184920 d[n_]:=n+Floor[n*r/u]+Floor[n*s/u]+Floor[n*t/u];
%t A184920 Table[a[n],{n,1,120}]  (* A184920 *)
%t A184920 Table[b[n],{n,1,120}]  (* A184921 *)
%t A184920 Table[c[n],{n,1,120}]  (* A184922 *)
%t A184920 Table[d[n],{n,1,120}]  (* A184923 *)
%Y A184920 Cf. A184912, A184921, A184922, A184923.
%K A184920 nonn
%O A184920 1,1
%A A184920 _Clark Kimberling_, Jan 26 2011