A014313 Numbers with exactly 5 ones in binary expansion.

31, 47, 55, 59, 61, 62, 79, 87, 91, 93, 94, 103, 107, 109, 110, 115, 117, 118, 121, 122, 124, 143, 151, 155, 157, 158, 167, 171, 173, 174, 179, 181, 182, 185, 186, 188, 199, 203, 205, 206, 211, 213, 214, 217, 218, 220, 227, 229, 230, 233, 234, 236, 241, 242

Offset: 1

Author

Al Black (gblack(AT)nol.net)

Keywords

• nonn
• base
• easy

Comments

Appears to give all n such that 4096 is the highest power of 2 dividing A005148(n). - Benoit Cloitre, Jun 22 2002

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015. See p. 18 for Mathematica code irwinSums.m.
• Index entries for 2-automatic sequences.

Crossrefs

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

Cf. A005148, A007088, A038447, A057168.

Programs

Haskell

a014313 = f . a038447 where
   f x = if x == 0 then 0 else 2 * f x' + b  where (x', b) = divMod x 10
-- Reinhard Zumkeller, Jan 06 2015

Mathematica

Select[ Range[31, 240], Total[IntegerDigits[#, 2]] == 5&]

PARI

sum_of_bits(n) = if(n<1, 0, sum_of_bits(floor(n/2))+n%2)
isA014313(n) = (sum_of_bits(n) == 5); \\ Michael B. Porter, Oct 21 2009

PARI

is(n)=hammingweight(n)==5 \\ Charles R Greathouse IV, Nov 17 2013

PARI

print1(t=2^5-1); for(i=2, 50, print1(", "t=A057168(t))) \\ M. F. Hasler, Aug 27 2014

Python

from itertools import islice
def A014313_gen(): # generator of terms
    yield (n:=31)
    while True: yield (n:=((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b)
A014313_list = list(islice(A014313_gen(),30)) # Chai Wah Wu, Mar 06 2025

Formula

a(n+1) = A057168(a(n)). - M. F. Hasler, Aug 27 2014

A038447(n) = A007088(a(n)). - Reinhard Zumkeller, Jan 06 2015

Sum_{n>=1} 1/a(n) = 1.390704528210321982529622080740025763242354253694629591331835888395977392151... (calculated using Baillie's irwinSums.m, see Links). - Amiram Eldar, Feb 14 2022

Extensions

Extension and program by Olivier Gérard