A242252 Start with n-th odd prime, and repeatedly subtract the greatest prime until either 0 or 1 remains. (The result is the "primes-greedy residue" of the n-th odd prime, which is "primes-greedy summable" if its residue = 0, as at A242255; see Comments.)

1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1

Offset

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 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 A242252, s = (2,3,5,7,11, ... ) = A000040.

Links

Clark Kimberling, Table of n, a(n) for n = 1..2000

Example

n ... n-th odd prime ... a(n)
1 ... 3 ................ 1 = 3 - 2
2 ... 5 ................ 0 = 5 - 3 - 2
3 ... 7 ................ 0 = 7 - 5 - 2
4 ... 11 ............... 1 = 11 - 7 - 3
5 ... 13 ............... 0 = 13 - 11 - 2
34 .. 149 .............. 1 = 149 - 139 - 7 - 2

Mathematica

z = 200;  s = Table[Prime[n], {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}]  (* A242252 *)
c = Table[Length[t[[n]][[2]]], {n, 2, z}] (* A242253 *)
f = 1 + Flatten[Position[tr, 0]]  (* A242254 *)
Prime[f]  (* A242255 *)
f1 = Prime[Complement[Range[Max[f]], f]] (* A242256 *)
(* Peter J. C. Moses, May 06 2014 *)

Crossrefs

Cf. A242253, A242254, A242255, A242256, A241833, A000040.

Sequence in context: A353380 A204171 A267612 * A100810 A373481 A174889

Adjacent sequences: A242249 A242250 A242251 * A242253 A242254 A242255

Keyword

nonn,easy

Author

Clark Kimberling, May 09 2014

Status

approved