A014312 Numbers with exactly 4 ones in binary expansion.

15, 23, 27, 29, 30, 39, 43, 45, 46, 51, 53, 54, 57, 58, 60, 71, 75, 77, 78, 83, 85, 86, 89, 90, 92, 99, 101, 102, 105, 106, 108, 113, 114, 116, 120, 135, 139, 141, 142, 147, 149, 150, 153, 154, 156, 163, 165, 166, 169, 170, 172, 177, 178, 180, 184, 195, 197

Offset

1,1

Links

T. D. Noe and Ivan Neretin, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

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.

Formula

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

a(n) = 2^A194882(n-1) + 2^A194883(n-1) + 2^A194884(n-1) + 2^A127324(n-1). - Ridouane Oudra, Sep 06 2020

Sum_{n>=1} 1/a(n) = 1.399770961748474333075618147113153558623203796657745865012742162098738541849... (calculated using Baillie's irwinSums.m, see Links). - Amiram Eldar, Feb 14 2022

Mathematica

Select[ Range[ 180 ], (Count[ IntegerDigits[ #, 2 ], 1 ]==4)& ] (* Olivier Gérard *)

Prog

(Perl) $N = 4;
my $vector = 2 ** $N - 1;  # first key (15)
for (1..100) {
  print "$vector, ";
  my ($v, $d) = ($vector, 0);
  until ($v & 1 or !$v) { $d = ($d << 1)|1; $v >>= 1 }
  $vector += $d + 1 + (($v ^ ($v + 1)) >> 2);  # next key
} # Ruud H.G. van Tol, Mar 02 2014
(PARI) for(n=0, 10^3, if(hammingweight(n)==4, print1(n, ", "))); \\ Joerg Arndt, Mar 04 2014
(PARI) print1(t=15); for(i=2, 50, print1(", "t=A057168(t))) \\ M. F. Hasler, Aug 27 2014
(Python)
A014312_list = [2**a+2**b+2**c+2**d for a in range(3, 6) for b in range(2, a) for c in range(1, b) for d in range(c)] # Chai Wah Wu, Jan 24 2021
(Python)
from itertools import islice
def A014312_gen(): # generator of terms
    yield (n:=15)
    while True: yield (n:=n^((a:=-n&n+1)|(a>>1)) if n&1 else ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b)
A014312_list = list(islice(A014312_gen(), 20)) # Chai Wah Wu, Mar 10 2025
(Rust)
pub const fn next_choice(value: usize) -> usize {
  // Passing a term will return the next number in the sequence
  let zeros = value.trailing_zeros();
  let ones = (value >> zeros).trailing_ones();
  value + (1 << zeros) + (1 << (ones - 1)) - 1
} // Andrew Bennett, Jan 07 2022

Crossrefs

Cf. A090706.

Cf. A000079, A018900, A014311, A014313, A023688, A023689, A023690, A023691 (Hamming weight = 1, 2, ..., 9), A057168.

Cf. A194882, A194883, A194884, A127324.

Sequence in context: A219683 A166665 A317295 * A371894 A325823 A129387

Adjacent sequences: A014309 A014310 A014311 * A014313 A014314 A014315

Keyword

nonn,base,easy,changed

Author

Al Black (gblack(AT)nol.net)

Extensions

Extension by Olivier Gérard

Status

approved