A023688 Numbers with exactly 6 ones in binary expansion.

63, 95, 111, 119, 123, 125, 126, 159, 175, 183, 187, 189, 190, 207, 215, 219, 221, 222, 231, 235, 237, 238, 243, 245, 246, 249, 250, 252, 287, 303, 311, 315, 317, 318, 335, 343, 347, 349, 350, 359, 363, 365, 366, 371, 373

Offset: 1

Author

Olivier Gérard

Keywords

• nonn
• base
• easy

Comments

Sequence appears to include all numbers m such that 8^5 is the highest power of 2 dividing A005148(m). General conjecture: numbers k such that 8^j is the highest power of 2 dividing A005148(k) is the same sequence as numbers having exactly (j+1) 1's in their binary representation. - Benoit Cloitre, Jun 22 2002

Links

• Ivan Neretin, 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.

Crossrefs

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

Cf. A005148, A057168.

Programs

Mathematica

Select[ Range[ 63, 380 ], (Count[ IntegerDigits[ #, 2 ], 1 ]==6)& ]

PARI

is_A023688(n)=hammingweight(n)==6 \\ M. F. Hasler, Aug 27 2014

PARI

print1(t=2^6-1); for(i=2, 50, print1(", "t=A057168(t))) \\ M. F. Hasler, Aug 27 2014

Python

from itertools import islice
def A023688_gen(): # generator of terms
    yield (n:=63)
    while True: yield (n:=((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b)
A023688_list = list(islice(A023688_gen(),30)) # Chai Wah Wu, Mar 06 2025

Formula

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

Sum_{n>=1} 1/a(n) = 1.387753111935705074750004158584017188750706394077047633137401652680870607884... (calculated using Baillie's irwinSums.m, see Links). - Amiram Eldar, Feb 14 2022