A321319 Smallest power of 2 obtainable by partitioning the binary representation of n into consecutive blocks and then summing.

1, 1, 2, 1, 2, 2, 4, 1, 2, 2, 4, 2, 4, 4, 4, 1, 2, 2, 4, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 4, 16, 1, 2, 2, 4, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 4, 8, 2, 4, 4, 4, 4, 4, 4, 8, 4, 4, 4, 8, 4, 8, 8, 8, 1, 2, 2, 4, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 4, 8, 2, 4, 4, 4, 4, 4, 4

Offset

1,3

Links

Robert Israel, Table of n, a(n) for n = 1..10000

E. Berlekamp and J. Buhler, Puzzle 6, Puzzles column, Emissary Fall (2011) 9.

Steve Butler, Ron Graham, and Richard Stong, Collapsing numbers in bases 2, 3, and beyond, in The Proceedings of the Gathering for Gardner 10 (2012).

Steve Butler, Ron Graham, and Richard Strong, Inserting plus signs and adding, Amer. Math. Monthly 123 (3) (2016), 274-279.

Example

For n = 13, we can partition its binary representation as follows (showing partition and sum of terms): (1101):13, (1)(101):6, (11)(01):4, (110)(1):7, (1)(1)(01):3, (1)(10)(1):4, (11)(0)(1):4, (1)(1)(0)(1):3. Thus the smallest power of 2 is 4.

Maple

g:= proc(n) option remember; local i, R;
     R:= {n};
     for i from 1 to ilog2(n) do
       R:= R union (procname(floor(n/2^i)) +~ (n mod 2^i));
     od;
     R;
end proc:
g(0):= {0}: g(1):= {1}:
f:= proc(n) local x;
  for x in sort(convert(g(n), list)) do
    if x = 2^padic:-ordp(x, 2) then
      return x
    fi
  od;
  -1
end proc:
map(f, [$1..100]); # Robert Israel, Jan 30 2026

Mathematica

sumInt[LL_] := Total[(FromDigits[#1, 2] & ) /@ LL];
BGS[{{1}, {1}, {1}, {1}, {1}}] = {{1}, {1, 1, 1, 1}};
BGS[{{1}, {1}, {1}, {1}, {1}, {0}}] = {{1, 1}, {1, 1}, {1, 0}};
BGS[L_ /; Total[Flatten[L]] == 5] := (p =
 Position[Range[Length[L]-2], x_Integer /; L[[x+{0, 1}]]==={{1}, {0}}, {1}, 1][[1, 1]];
   Join[L[[1 ;; p - 1]], {L[[p + {0, 1, 2}, 1]]},
    L[[p + 3 ;; Length[L]]]]);
tripleMerge[L_, pow_]:=(
      p=Position[Range[Length[L]-2], x_Integer/; L[[x]]==={1}, {1}, 1];
   If[p === {}, L,
   p = p[[1, 1]]; J = Join[L[[1 ;; p - 1]], {L[[p + {0, 1, 2}, 1]]},
      L[[p+3;; Length[L]]]]; If[sumInt[J] <= pow, J, L]]);
doubleMerge[L_, pow_] := (
   p=Position[Range[Length[L]-1], x_Integer/; L[[x]] === {1}, {1}, 1];
   If[p === {}, L, p = p[[1, 1]];
     J = Join[L[[1 ;; p - 1]], {L[[p + {0, 1}, 1]]},
     L[[p + 2 ;; Length[L]]]]; If[sumInt[J] <= pow, J, L]]);
BGS[L_ /; Total[Flatten[L]] != 5] := (old = L;
   pow = 2^Ceiling[Log2[Total[Flatten[L]]]];
   While[new = tripleMerge[old, pow]; new =!= old, old = new];
       While[new = doubleMerge[old, pow]; new =!= old, old = new];
   new);
sumInt[BGS[List /@ IntegerDigits[#, 2]]] & /@ Range[86] (* Stan Wagon, Jan 25 2026 *)

Crossrefs

Cf. A321318, A321320, A321321.

Sequence in context: A367970 A391070 A345161 * A048896 A130831 A151678

Adjacent sequences: A321316 A321317 A321318 * A321320 A321321 A321322

Keyword

nonn

Author

Jeffrey Shallit, Nov 04 2018

Status

approved