A392117 Lower Wythoff partition sequence of the primes; see Comments.

2, 3, 11, 13, 19, 29, 37, 41, 47, 53, 67, 73, 79, 89, 97, 103, 107, 109, 131, 139, 151, 163, 179, 181, 191, 193, 197, 223, 229, 233, 239, 241, 257, 269, 277, 281, 283, 313, 317, 337, 353, 359, 379, 383, 389, 401, 419, 421, 439, 449, 457, 461, 463, 467, 491

Offset

1,1

Comments

Let s = (s(n)) be an increasing sequence of positive integers, and define sequences u = (u(n)) and v = (v(n)) as follows:

u(1) = s(1);

v(1) = least term of s that is >= u(1) + s(1);

u(n) = least term of s that is not in {u(k) : k < n} and not in {v(k) : k < n};

v(n) = least unused term of s that is greater than u(n) + s(n), for n >= 1.

Then u is the lower Wythoff partition sequence of s, and v is the upper Wythoff partition sequence of s.

If s = A000027, then u = A000201 and v = A001950, these being the classical lower and upper Wythoff sequences.

Links

Table of n, a(n) for n=1..55.

Mathematica

w[s_List] := Module[{u = {}, v = {}, used = <||>, n = 1, i = 1, target, vt},
  While[i <= Length@s, While[i <= Length@s && KeyExistsQ[used, i], i++];
   If[i > Length@s, Break[]]; AppendTo[u, s[[i]]]; used[i] = True;
   target = u[[-1]] + s[[n]]; vt = -1; Do[If[! KeyExistsQ[used, j] && s[[j]] >= target, vt = j; Break[]], {j, 1, Length@s}]; If[vt == -1, u = Drop[u, -1]; Break[], AppendTo[v, s[[vt]]]; used[vt] = True; n++]; ]; {u, v}]
s = Prime[Range[200]]; (* User: put your increasing sequence s here *)
u = First[w[s]]  (* lower Wythoff partition sequence of s *)
v = Last[w[s]]  (* upper Wythoff partition sequence of s *)
(* Peter J. C. Moses, Dec 23 2025 *)

Crossrefs

Cf. A000040, A000201, A001950, A392118, A392119, A392121.

Sequence in context: A386351 A038967 A386090 * A001916 A089151 A115669

Adjacent sequences: A392114 A392115 A392116 * A392118 A392119 A392120

Keyword

nonn

Author

Clark Kimberling, Jan 19 2026

Status

approved