A292895 a(n) is the least positive k such that the Hamming weight of k equals the Hamming weight of k + n.

1, 1, 2, 1, 4, 5, 2, 1, 8, 3, 10, 6, 4, 5, 2, 1, 16, 3, 6, 5, 20, 3, 12, 10, 8, 9, 10, 6, 4, 5, 2, 1, 32, 3, 6, 5, 12, 3, 10, 9, 40, 11, 6, 5, 24, 3, 20, 18, 16, 7, 18, 17, 20, 12, 12, 10, 8, 9, 10, 6, 4, 5, 2, 1, 64, 3, 6, 5, 12, 3, 10, 9, 24, 11, 6, 5, 20, 3, 18, 17, 80, 7, 22, 14, 12, 13, 10, 9, 48

Offset

0,3

Comments

Inspired by A292849.

The Hamming weight of a number n is given by A000120(n).

Let b(n) be the smallest t such that a(t) = n. Initial values of b(n) are 0, 2, 9, 4, 5, 11, 49, 8, 25, 10, 41, 22, 85, 83, 225, 16, 51, 47, 177, 20, ... See the logarithmic line graph of first 10^3 terms of b(n) sequence in Links section.

Apparently, n = a(n) iff n belongs to A094958. - Rémy Sigrist, Oct 02 2017

Links

Rémy Sigrist, Table of n, a(n) for n = 0..16384

Altug Alkan, A logarithmic scatterplot of a(n) for n <= 10^6

Altug Alkan, A logarithmic line graph of b(n) for n <= 10^3

Formula

a(n) <= n for n >= 1.

a(2*n) = 2*a(n) for n >= 1.

a(2^m) = 2^m and a(5*2^m) = 5*2^m for m >= 0.

a(2^m - 1) = 1 for m >= 0.

a(2^m + 1) = 3 and a(2^m - 3) = 5 for m >= 3.

a(2^m + 3) = 5 for m >= 4.

a((2^m - 1)^2) = 2^m - 1 for m >= 1.

a(2^(m + 2) + 2^m - 1) = 2^m + 1 m >= 1.

a((2^m + 1)^2) = 7 for m >= 3.

Example

a(49) = 7 since A000120(7) = A000120(7 + 49) and 7 is the least number with this property.

Maple

N:= 1000: # to get all terms before the first where n+a(n)>N
H:= Array(0..N, t -> convert(convert(t, base, 2), `+`)):
f:= proc(n) local k;
  for k from 1 to N-n do
    if H[k]=H[k+n] then return k fi
  od:
0
end proc:
R:= NULL:
for n from 0 do
v:= f(n);
if v = 0 then break fi;
  R:= R, v;
od:
R; # Robert Israel, Sep 27 2017

Mathematica

h[n_] := First@ DigitCount[n, 2]; a[n_] := Block[{k=1}, While[h[k] != h[k + n], k++]; k]; Array[a, 90] (* Giovanni Resta, Sep 28 2017 *)

Prog

(PARI) a(n) = {my(k=1); while ((hammingweight(k)) != hammingweight(n+k), k++); k; }

Crossrefs

Cf. A000120, A094958, A292849.

Sequence in context: A072436 A247503 A343430 * A371015 A090077 A229763

Adjacent sequences: A292892 A292893 A292894 * A292896 A292897 A292898

Keyword

nonn,base,easy,look

Author

Rémy Sigrist and Altug Alkan, Sep 26 2017

Status

approved