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

Author

Marc LeBrun, Sep 14 2000

Keywords

• easy
• nonn
• nice

Comments

Binary weight is given by A000120.

Examples

			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).
		

References

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

Links

• 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)

Crossrefs

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).

Programs

Haskell

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

Mathematica

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 *)

PARI

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

PARI

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

Python

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

Python

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

Formula

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