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A278103 Irregular triangle T(n,k) := A278101(n,k) for k = 1..A278102(n), read by rows.

%I A278103 #14 Dec 12 2016 09:15:12
%S A278103 1,4,2,9,8,3,16,8,25,18,12,36,32,27,20,49,32,64,50,48,45,81,72,100,98,
%T A278103 75,121,98,144,128,108,169,162,147,125,196,162,225,200,192,180,256,
%U A278103 242,289,288,243,324,288,361,338,300,400,392,363,320,441,392,484,450,432
%N A278103 Irregular triangle T(n,k) := A278101(n,k) for k = 1..A278102(n), read by rows.
%C A278103 Each row is the longest strictly decreasing prefix of the corresponding row of A278101.
%D A278103 R. B. Eggleton, J. S. Kimberley and J. A. MacDougall, Square-free rank of integers, submitted.
%H A278103 Jason Kimberley, <a href="/A278103/b278103.txt">Table of n, a(n) for n = 1..10001 (the first 3304 rows of the triangle)</a>
%F A278103 T(n,k) = A278104(n,k) * A005117(k) where this triangle and A278104 both have row length sequence A278102.
%e A278103 The first 23 rows are:
%e A278103 1;
%e A278103 4, 2;
%e A278103 9, 8, 3;
%e A278103 16, 8;
%e A278103 25, 18, 12;
%e A278103 36, 32, 27, 20;
%e A278103 49, 32;
%e A278103 64, 50, 48, 45;
%e A278103 81, 72;
%e A278103 100, 98, 75;
%e A278103 121, 98;
%e A278103 144, 128, 108;
%e A278103 169, 162, 147, 125;
%e A278103 196, 162;
%e A278103 225, 200, 192, 180;
%e A278103 256, 242;
%e A278103 289, 288, 243;
%e A278103 324, 288;
%e A278103 361, 338, 300;
%e A278103 400, 392, 363, 320;
%e A278103 441, 392;
%e A278103 484, 450, 432, 405, 384;
%e A278103 529, 512, 507, 500, 486, 448;
%t A278103 Map[TakeWhile[FoldList[Function[s, Boole[s < 0] #2][#2 - #1] &, #], # > 0 &] &, #] &@ Map[DeleteCases[#, 0] &, Table[Boole[SquareFreeQ@ k] k Floor[n/Sqrt@ k]^2, {n, 23}, {k, n^2}] ] // Flatten (* _Michael De Vlieger_, Nov 24 2016 *)
%o A278103 (Magma)
%o A278103 A277647:=func<n,k|Isqrt(n^2 div k)>;
%o A278103 A278101_row:=func<n|[a^2*k where a is A277647(n,k):k in[1..n^2]|IsSquarefree(k)]>;
%o A278103 A278103_row:=func<n|(exists(dec){row[1..j]:j in[1..#row-1]|row[j]le row[j+1]}select dec else[1]) where row is A278101_row(n) >;
%o A278103 &cat[A278103_row(n):n in[1..23]];
%Y A278103 Cf. A005117, A278101, A278102, A278104.
%K A278103 nonn,tabf,easy
%O A278103 1,2
%A A278103 _Jason Kimberley_, Nov 15 2016