

A242293


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


4



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

2,1


COMMENTS

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 sgreedy residue of x, and call s(i(1)) + ... + s(i(k)) the sgreedy sum for x. If r = 0, call x sgreedy 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, ...).


LINKS

Clark Kimberling, Table of n, a(n) for n = 2..2000


EXAMPLE

n ... n^3 ... a(n)
1 ... 1 .... (undefined)
2 ... 8 ..... 7 = 8  1
3 ... 27 .... 18 = 27  8  1
4 ... 64 .... 28 = 64  27  8  1
5 ... 125 ... 25 = 125  64  27  8  1
6 ... 216 ... 0 = 216  125  64  27
7 ... 343 ... 1 = 343  216  125  1
8 ... 512 ... 8 = 512  125  27  8  1
9 ... 729 ... 0 = 729  512  216  1


MATHEMATICA

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}]
r[n_] := s[[n]]  Total[t[[n]][[2]]];
tr = Table[r[n], {n, 2, z}] (* A242293 *)
c = Table[Length[t[[n]][[2]]], {n, 2, z}] (* A242294 *)
f = 1 + Flatten[Position[tr, 0]] (* A242295*)
f^3 (* A242296 *) (* Peter J. C. Moses, May 06 2014 *)


CROSSREFS

Cf. A242294, A242295, A242296, A241833, A242284, A000578.
Sequence in context: A090098 A101865 A138391 * A179298 A144175 A017473
Adjacent sequences: A242290 A242291 A242292 * A242294 A242295 A242296


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, May 10 2014


STATUS

approved



