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%I #14 Sep 25 2021 03:42:51
%S 3,4,6,7,9,10,12,13,14,15,17,18,19,21,22,23,24,25,27,28,29,30,31,33,
%T 34,35,36,37,39,40,41,42,43,44,45,47,48,49,50,51,52,54,55,56,57,58,59,
%U 60,61,63,64,65,66,67,68,69,70,72,73,74,75,76,77,78,79,81,82,83,84,85,86,87,88,89,90,92,93,94,95,96,97,98,99,100,102,103,104,105,106,107,108,109,110,111,112,114,115,116,117,118,119,120,121,122,123,124,126,127,128,129,130,131,132,133,134,135,136,138,139,140
%N Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=5i (A008587) and g(j)=j-th pentagonal number (A000326). Complement of A186496.
%e First, write
%e ...5..10..15..20..25..30..35..40... (5i),
%e 1..5......12......22............35..(pentagonal numbers).
%e Then replace each number by its rank, where ties are settled by ranking 5i after the pentagonal number:
%e a=(3,4,6,7,9,10,12,13,14,15,17,...)=A186495,
%e b=(1,2,5,8,11,16,20,26,32,38,46,..)=A186496.
%t (* Adjusted joint rank sequences a and b, using general formula for ranking 1st degree u*n+v and 2nd degree x*n^2+y*n+z. *)
%t d=-1/2; u=5; v=0; x=3/2; y=-1/2;
%t h[n_]:=(-y+(4x(u*n+v-d)+y^2)^(1/2))/(2x);
%t a[n_]:=n+Floor[h[n]];
%t k[n_]:=(x*n^2+y*n-v+d)/u;
%t b[n_]:=n+Floor[k[n]];
%t Table[a[n],{n,1,120}] (* A186495 *)
%t Table[b[n],{n,1,100}] (* A186496 *)
%Y Cf. A000326, A008587, A186350, A186493, A186494, A186496.
%K nonn
%O 1,1
%A _Clark Kimberling_, Feb 22 2011
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