A242293 - OEIS

%I #5 May 15 2014 10:16:59

%S 7,18,28,25,0,1,8,0,19,15,18,0,0,19,11,15,0,0,7,9,20,27,10,0,6,0,0,15,

%T 6,11,8,9,11,6,27,10,23,0,0,0,2,2,0,9,0,9,10,0,15,27,6,17,2,21,16,0,

%U 12,5,1,17,26,6,18,6,2,0,10,1,2,14,21,10,5,17,11

%N Greedy residue sequence of cubes 2^3, 3^3, 4^3, ...

%C Suppose that s = (s(1), s(2), ... ) is a sequence of real numbers such that for every real number u, at most finitely many s(i) are < u, and suppose that x > min(s).  We shall apply the greedy algorithm to x, using terms of s.  Specifically, let i(1) be an index i such that s(i) = max{s(j) < x}, and put d(1) = x - s(i(1)).  If d(1) < s(i) for all i, put r = x - s(i(1)).  Otherwise, let i(2) be an index i such that s(i) = max{s(j) < x - s(i(1))}, and put d(2) = x - s(i(1)) - s(i(2)).  If d(2) < s(i) for all i, put r = x - s(i(1)) - s(i(2)).  Otherwise, let i(3) be an index i such that s(i) = max{s(j) < x - s(i(1)) - s(i(2))}, and put d(3) = x - s(i(1)) - s(i(2)) - s(i(3)).  Continue until reaching k such that d(k) < s(i) for every i, and put r = x - s(i(1)) - ... - s(i(k)).  Call r the s-greedy residue of x, and call s(i(1)) + ... + s(i(k)) the s-greedy sum for x.   If r = 0, call x s-greedy summable.  If s(1) = min(s) < s(2), then taking x = s(i) successively for i = 2, 3,... gives a residue r(i) for each i; call (r(i)) the greedy residue sequence for s.  When s is understood from context, the prefix "s-" is omitted.  For A242293, s = (1^3, 2^3, 3^3, ...).

%H Clark Kimberling, <a href="/A242293/b242293.txt">Table of n, a(n) for n = 2..2000</a>

%e n ... n^3 ... a(n)

%e 1 ... 1 .... (undefined)

%e 2 ... 8 ..... 7 = 8 - 1

%e 3 ... 27 .... 18 = 27 - 8 - 1

%e 4 ... 64 .... 28 = 64 - 27 - 8 - 1

%e 5 ... 125 ... 25 = 125 - 64 - 27 - 8 - 1

%e 6 ... 216 ... 0 = 216 - 125 - 64 - 27

%e 7 ... 343 ... 1 = 343 - 216 - 125 - 1

%e 8 ... 512 ... 8 = 512 - 125 - 27 - 8 - 1

%e 9 ... 729 ... 0 = 729 - 512 - 216 - 1

%t z = 200;  s = Table[n^3, {n, 1, z}]; t = Table[{s[[n]], #, Total[#] == s[[n]]} &[DeleteCases[-Differences[FoldList[If[#1 - #2 >= 0, #1 - #2, #1] &, s[[n]], Reverse[Select[s, # < s[[n]] &]]]], 0]], {n, z}]

%t r[n_] := s[[n]] - Total[t[[n]][[2]]];

%t tr = Table[r[n], {n, 2, z}]  (* A242293 *)

%t c = Table[Length[t[[n]][[2]]], {n, 2, z}] (* A242294 *)

%t f = 1 + Flatten[Position[tr, 0]]  (* A242295*)

%t f^3  (* A242296 *) (* _Peter J. C. Moses_, May 06 2014 *)

%Y Cf. A242294, A242295, A242296, A241833, A242284, A000578.

%K nonn,easy

%O 2,1

%A _Clark Kimberling_, May 10 2014