A171942 -id:A171942 - cp's OEIS frontend

A057168 Next larger integer with same binary weight (number of 1 bits) as n.

2, 4, 5, 8, 6, 9, 11, 16, 10, 12, 13, 17, 14, 19, 23, 32, 18, 20, 21, 24, 22, 25, 27, 33, 26, 28, 29, 35, 30, 39, 47, 64, 34, 36, 37, 40, 38, 41, 43, 48, 42, 44, 45, 49, 46, 51, 55, 65, 50, 52, 53, 56, 54, 57, 59, 67, 58, 60, 61, 71, 62, 79, 95, 128, 66, 68, 69, 72, 70, 73, 75

Offset: 1

Marc LeBrun, Sep 14 2000

Binary weight is given by A000120.

			a(6)=9 since 6 has two one-bits (i.e., 6=2+4) and 9 is the next higher integer of binary weight two (7 is weight three and 8 is weight one).

Donald Knuth, The Art of Computer Programming, Vol. 4A, section 7.1.3, exercises 20-21.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. Beeler, R. W. Gosper, and R. Schroeppel, HAKMEM, MIT Artificial Intelligence Laboratory, Memo AIM-239, February 1972, Item 175 by Gosper, page 81. Also HTML transcription.
Donald E. Knuth, The Art of Computer Programming, Pre-Fascicle 1A, Draft of Section 7.1.3, 2008. Exercises 20 and 21 page 54 and answers pages 75-76.
Marc B. Reynolds, Popcount walks: next, previous, toward and nearest (2023)

Cf. A000120, A006519, A057169, A000051, A052548, A140504, A000225, A055010, A007283, A171942.

Cf. A000079, A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691 (Hamming weight = 1, 2, ..., 9).

a057168 n = a057168_list !! (n-1)
a057168_list = f 2 $ tail a000120_list where
   f x (z:zs) = (x + length (takeWhile (/= z) zs)) : f (x + 1) zs
-- Reinhard Zumkeller, Aug 26 2012

a[n_] := (bw = DigitCount[n, 2, 1]; k = n+1; While[ DigitCount[k, 2, 1] != bw, k++]; k); Table[a[n], {n, 1, 71}](* Jean-François Alcover, Nov 28 2011 *)

a(n)=my(u=bitand(n,-n),v=u+n);(bitxor(v,n)/u)>>2+v \\ Charles R Greathouse IV, Oct 28 2009

A057168(n)=n+bitxor(n,n+n=bitand(n,-n))\n\4+n \\ M. F. Hasler, Aug 27 2014

def a(n): u = n&-n; v = u+n; return (((v^n)//u)>>2)+v
print([a(n) for n in range(1, 72)]) # Michael S. Branicky, Jul 10 2022 after Charles R Greathouse IV

def A057168(n): return ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b # Chai Wah Wu, Mar 06 2025

From Reinhard Zumkeller, Aug 18 2008: (Start)
a(A000079(n)) = A000079(n+1);
a(A000051(n)) = A052548(n);
a(A052548(n)) = A140504(n);
a(A000225(n)) = A055010(n);
a(A007283(n)) = A000051(n+2). (End)
a(n) = MIN{m: A000120(m)=A000120(n) and m>n}. - Reinhard Zumkeller, Aug 15 2009
For k,m>0, a((2^k-1)*2^m) = 2^(k+m)+2^(k-1)-1. - Chai Wah Wu, Mar 07 2025
If n is odd, then a(n) = XOR(n,OR(a,a/2)) where a = AND(-n,n+1). - Chai Wah Wu, Mar 08 2025

A171898 Forward van Eck transform of A181391.

Original entry on oeis.org

1, 2, 6, 2, 2, 5, 1, 6, 42, 5, 2, 4, 5, 9, 14, 3, 9, 3, 15, 2, 4, 6, 17, 3, 6, 32, 56, 5, 3, 131, 5, 11, 5, 3, 20, 6, 2, 8, 15, 31, 170, 3, 31, 18, 3, 3, 33, 5, 1, 11, 46, 56, 4, 37, 152, 307, 3, 7, 92, 4, 7, 62, 52, 3, 42, 3, 6, 2, 19, 6, 8, 3, 9, 3, 650, 2, 23, 8, 223, 7, 206, 3, 21, 25, 5, 8

Offset: 1

N. J. A. Sloane, Oct 22 2010

nonn

Given a sequence a, the forward van Eck transform b is defined as follows: If a(n) also appears again in a later position, let a(m) be the next occurrence, and set b(n)=m-n; otherwise b(n)=0.
This is a permutation of the positive terms in A181391, where each term m > 0 from that sequence is shifted backwards m+1 positions. - Jan Ritsema van Eck, Aug 16 2019
The backwards van Eck transform searches backwards for a repeated value: if a(n) also has appeared in earlier positions, a(m)=a(n) with mR. J. Mathar, Jun 24 2021

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A181391 (van Eck's sequence), A171899, A171942.

ECKf:=proc(a) local b,i,m,n;
if whattype(a) <> list then RETURN([]); fi:
b:=[];
for n from 1 to nops(a)-1 do
# does a(n) appear again?
m:=0;
for i from n+1 to nops(a) do
if (a[i]=a[n]) then m:=i-n; break; fi
od:
b:=[op(b),m];
od:
b:=[op(b),0];
RETURN(b);
end:

terms = 100;
m = 14 terms; (* Increase m until no zero appears in the output *)
ClearAll[b, last]; b[] = 0; last[] = -1; last[0] = 2; nxt = 1;
Do[hist = last[nxt]; b[n] = nxt; last[nxt] = n; nxt = 0; If[hist > 0, nxt = n - hist], {n, 3, m}];
A181391 = Array[b, m];
ECKf[a_List] := Module[{b = {}, i, m, n}, For[n = 1, n <= Length[a]-1, n++, m = 0; For[i = n+1, i <= Length[a], i++, If[a[[i]] == a[[n]], m = i-n; Break[]]]; b = Append[b, m]]; b = Append[b, 0]; Return[b]];
ECKf[A181391][[;; terms]] (* Jean-François Alcover, Oct 30 2020, after Maple *)

From Jan Ritsema van Eck, Aug 16 2019: (Start)
A181391(i+a(i)+1) = a(i) for any i, a(i)>0.
Conversely, a(j-A181391(j)-1) = A181391(j) for any j, A181391(j)>0. (End)

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A057168 Next larger integer with same binary weight (number of 1 bits) as n.

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