A275288 -id:A275288 - cp's OEIS frontend

A272036 Numbers n such that the sum of the inverse of the exponents in the binary expansion of 2n is equal to 1.

1, 38, 2090, 16902, 18954, 18988, 131334, 133386, 133420, 148258, 150284, 524314, 524348, 526386, 541212, 543250, 543284, 655644, 657682, 657716, 672568, 674580, 8388742, 8390794, 8390828, 8405666, 8407692, 8520098, 8522124, 8536962, 8536996, 8539048, 8913052, 8915090

Offset: 1

Michel Marcus, Apr 18 2016

nonn

That is, numbers such that both A116416(n) and A116417(n) are equal to 1.
Intersection of A272034 and A272035.
A number m with an exponent k in the binary sum must have another power of 2 having an exponent at least A275288(k). - David A. Corneth, Apr 01 2017

			For n=38, 2*38_10 = 2^6 + 2^3 + 2^2 = 1001100_2, and 1/2 + 1/3 + 1/6 = 1.

Robert Israel, Table of n, a(n) for n = 1..1655 (first 200 terms from Peter Kagey)

Cf. A116416, A116417, A272034, A272035, A272083, A275288.

Select[Range[2^20], Total[1/Flatten@ Position[Reverse@ IntegerDigits[#, 2], 1]] == 1 &] (* Michael De Vlieger, Apr 18 2016 *)

is(n) = my(b = Vecrev(binary(n))); sum(k=1, #b, b[k]/k) == 1;

A359506 a(n) is the least integer m such that there exists a strictly increasing integer sequence n = b_1 < b_2 < ... < b_t = m with the property that b_1 XOR b_2 XOR ... XOR b_t = 0.

Original entry on oeis.org

0, 3, 5, 6, 7, 10, 9, 12, 11, 14, 13, 20, 15, 18, 17, 24, 19, 22, 21, 28, 23, 26, 25, 40, 27, 30, 29, 36, 31, 34, 33, 48, 35, 38, 37, 44, 39, 42, 41, 56, 43, 46, 45, 52, 47, 50, 49, 80, 51, 54, 53, 60, 55, 58, 57, 72, 59, 62, 61, 68, 63, 66, 65, 96, 67

Offset: 0

Peter Kagey, Jan 03 2023

nonn

XOR is the bitwise XOR function, A003987.
This sequence is a bijection from the nonnegative integers to A057716, the nonpowers of 2.
Let's call the sequences mentioned in the definition as "zero-XOR sequences", and their last terms as "enders". a(n) is then the least possible ender for any zero-XOR sequence starting with n. - Antti Karttunen, Nov 25 2024

			For n = 19, a(19) = 28 with the sequence 19 XOR 20 XOR 27 XOR 28 = 0.
A table illustrating the first eleven terms:
   n |a(n)| sequence
  ---+----+-------------------
   0 |  0 |  0
   1 |  3 |  1 XOR  2 XOR  3
   2 |  5 |  2 XOR  3 XOR  4 XOR  5
   3 |  6 |  3 XOR  5 XOR  6
   4 |  7 |  4 XOR  5 XOR  6 XOR  7
   5 | 10 |  5 XOR  6 XOR  9 XOR 10
   6 |  9 |  6 XOR  7 XOR  8 XOR  9
   7 | 12 |  7 XOR 11 XOR 12
   8 | 11 |  8 XOR  9 XOR 10 XOR 11
   9 | 14 |  9 XOR 10 XOR 13 XOR 14
  10 | 13 | 10 XOR 11 XOR 12 XOR 13

Antti Karttunen, Table of n, a(n) for n = 0..65537 (first 1000 terms from Peter Kagey)
Peter Kagey, Proof of bijection onto A057716 Note: there is a typo in this first revision of the proof. In the definition of f (which is now A378212) "Let f(n) be the least nonnegative integer k such that ...", the "least" should actually be "greatest", _Antti Karttunen_, Dec 01 2024, as communicated by _Peter Kagey_

Cf. A003987, A057716, A359507, A359508, A378212 (a left inverse).

Cf. A006255, A275288, A277278, A277494, A300516, A329732 (variants of the theme).

f:= proc(n) local k,S;
    S:= {n};
    for k from n+1 do
      S:= S union map(Bits:-Xor,S,k);
      if member(0,S) then return k fi;
    od;
end proc:
f(0):= 0:
map(f, [$0..100]); # Robert Israel, Jan 12 2023

f[n_] := Module[{k, S}, S = {n}; For[k = n+1, True, k++, S = S  ~Union~ BitXor[S, k]; If[MemberQ[S, 0], Return[k]]]];
f[0] = 0;
f /@ Range[0, 100] (* Jean-François Alcover, Jan 22 2023, after Robert Israel *)

a(n)= if (n==0, return (0), my (x=[n],y); for (m=n+1, oo, if (vecmin(y=[bitxor(v,m) | v<-x])==0, return (m), x=setunion(x,Set(y)))))  \\ Rémy Sigrist, Jan 12 2023

For n > 1, a(n) >= n + 3. a(4n) = 4n + 3 for n > 0. Conjecture: a(n) <= 5(n+1)/3. - Charles R Greathouse IV, Jan 12 2023

For all n >= 0, A378212(a(n)) = n. - Antti Karttunen, Nov 25 2024

cp's OEIS Frontend

A272036 Numbers n such that the sum of the inverse of the exponents in the binary expansion of 2n is equal to 1.

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