A043569 Numbers whose base-2 representation has exactly 2 runs.

2, 4, 6, 8, 12, 14, 16, 24, 28, 30, 32, 48, 56, 60, 62, 64, 96, 112, 120, 124, 126, 128, 192, 224, 240, 248, 252, 254, 256, 384, 448, 480, 496, 504, 508, 510, 512, 768, 896, 960, 992, 1008, 1016, 1020, 1022, 1024, 1536, 1792, 1920, 1984, 2016, 2032, 2040, 2044

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• base

Comments

Numbers whose binary representation contains the bit string "10" but not "01". Subsequence of A062289; set difference A062289 minus A101082. - Rick L. Shepherd, Nov 29 2004

Mersenne numbers (A000225) times powers of 2 (A000079). Therefore this sequence contains the even perfect numbers (A000396). - Alonso del Arte, Apr 21 2006

Links

• Lei Zhou, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A004736, A007814, A062289, A065442, A101082.

Programs

Maple

a:=proc(n) local nn,nd: nn:=convert(n,base,2): nd:={seq(nn[j]-nn[j-1],j=2..nops(nn))}: if n=2 then 2 elif nd={0,1} then n else fi end: seq(a(n),n=1..2100); # Emeric Deutsch, Apr 21 2006

Mathematica

Take[Sort[Flatten[Table[(2^x - 1)*(2^y), {x, 32}, {y, 32}]]], 54] (* Alonso del Arte, Apr 21 2006 *)
Select[Range[2500],Length[Split[IntegerDigits[#,2]]]==2&] (* or *) Select[Range[2500],SequenceCount[IntegerDigits[#,2],{1,0}]>0 && SequenceCount[ IntegerDigits[#,2],{0,1}]==0&] (* Harvey P. Dale, Oct 04 2024 *)

Python

def ok(n): b = bin(n)[2:]; return "10" in b and "01" not in b
print([m for m in range(2045) if ok(m)]) # Michael S. Branicky, Feb 04 2021

Python

def a_next(a_n): t = a_n >> 1; return (a_n | t) + (t & 1)
a_n = 2; a = []
for i in range(54): a.append(a_n); a_n = a_next(a_n) # Falk Hüffner, Feb 19 2022

Formula

This sequence is twice A023758. - Franklin T. Adams-Watters, Apr 21 2006

Sum_{n>=1} 1/a(n) = A065442. - Amiram Eldar, Feb 20 2022

A007814(a(n)) = A004736(n). - Lorenzo Sauras Altuzarra, Feb 01 2023

Extensions

More terms from Rick L. Shepherd, Nov 29 2004