|
%I #9 May 15 2014 10:13:28
%S 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,
%T 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,
%U 0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,1,1,1
%N 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.)
%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 A242252, s = (2,3,5,7,11, ... ) = A000040.
%H Clark Kimberling, <a href="/A242252/b242252.txt">Table of n, a(n) for n = 1..2000</a>
%e n ... n-th odd prime ... a(n)
%e 1 ... 3 ................ 1 = 3 - 2
%e 2 ... 5 ................ 0 = 5 - 3 - 2
%e 3 ... 7 ................ 0 = 7 - 5 - 2
%e 4 ... 11 ............... 1 = 11 - 7 - 3
%e 5 ... 13 ............... 0 = 13 - 11 - 2
%e 34 .. 149 .............. 1 = 149 - 139 - 7 - 2
%t 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 *)
%t c = Table[Length[t[[n]][[2]]], {n, 2, z}] (* A242253 *)
%t f = 1 + Flatten[Position[tr, 0]] (* A242254 *)
%t Prime[f] (* A242255 *)
%t f1 = Prime[Complement[Range[Max[f]], f]] (* A242256 *)
%t (* _Peter J. C. Moses_, May 06 2014 *)
%Y Cf. A242253, A242254, A242255, A242256, A241833, A000040.
%K nonn,easy
%O 1
%A _Clark Kimberling_, May 09 2014
|
