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%I #11 Dec 12 2016 09:15:45
%S 1,2,1,3,2,1,4,2,5,3,2,6,4,3,2,7,4,8,5,4,3,9,6,10,7,5,11,7,12,8,6,13,
%T 9,7,5,14,9,15,10,8,6,16,11,17,12,9,18,12,19,13,10,20,14,11,8,21,14,
%U 22,15,12,9,8,23,16,13,10,9,8,24,16,13,10,9,25,17,26,18,27,19,15,28,19
%N Irregular triangle T(n,k) := A277648(n,k) for k = 1...A278102(n), read by rows.
%C This triangle lists the "descending sequences across ranks" of Eggleton et al.
%D R. B. Eggleton, J. S. Kimberley and J. A. MacDougall, Square-free rank of integers, submitted.
%H Jason Kimberley, <a href="/A278104/b278104.txt">Table of n, a(n) for n = 1..10001 (the first 3304 rows of the triangle)</a>
%e The first 23 rows are:
%e 1;
%e 2, 1;
%e 3, 2, 1;
%e 4, 2;
%e 5, 3, 2;
%e 6, 4, 3, 2;
%e 7, 4;
%e 8, 5, 4, 3;
%e 9, 6;
%e 10, 7, 5;
%e 11, 7;
%e 12, 8, 6;
%e 13, 9, 7, 5;
%e 14, 9;
%e 15, 10, 8, 6;
%e 16, 11;
%e 17, 12, 9;
%e 18, 12;
%e 19, 13, 10;
%e 20, 14, 11, 8;
%e 21, 14;
%e 22, 15, 12, 9, 8;
%e 23, 16, 13, 10, 9, 8;
%t Map[Last, #, {2}] &@ Map[TakeWhile[FoldList[Function[s, Boole[s < 0] {First@ #2, Last@ #2}][First@ #2 - First@ #1] &, #], Total@ # > 0 &] &, #] &@ Map[DeleteCases[#, {0, 0}] &, Table[Boole[SquareFreeQ@ k] {k #^2, #} &@ Floor[n/Sqrt@ k], {n, 32}, {k, n^2}] ] // Flatten (* _Michael De Vlieger_, Nov 24 2016 *)
%o (Magma)
%o A277647:=func<n,k|Isqrt(n^2 div k)>;
%o A277648_row:=func<n|[A277647(n,k):k in[1..n^2]|IsSquarefree(k)]>;
%o A278101_row:=func<n|[A277647(n,k)^2*k:k in[1..n^2]|IsSquarefree(k)]>;
%o A278104_row:=func<n|(exists(dec){A277648_row(n)[1..j]:j in[1..#row-1]|row[j]le row[j+1]}select dec else[1]) where row is A278101_row(n) >;
%o &cat[A278104_row(n):n in[1..23]];
%K nonn,tabf,easy
%O 1,2
%A _Jason Kimberley_, Nov 15 2016
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