A014312 - OEIS

%I #77 Feb 24 2023 21:48:23

%S 15,23,27,29,30,39,43,45,46,51,53,54,57,58,60,71,75,77,78,83,85,86,89,

%T 90,92,99,101,102,105,106,108,113,114,116,120,135,139,141,142,147,149,

%U 150,153,154,156,163,165,166,169,170,172,177,178,180,184,195,197

%N Numbers with exactly 4 ones in binary expansion.

%H T. D. Noe and Ivan Neretin, <a href="/A014312/b014312.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)

%H Robert Baillie, <a href="http://arxiv.org/abs/0806.4410">Summing the curious series of Kempner and Irwin</a>, arXiv:0806.4410 [math.CA], 2008-2015. See p. 18 for Mathematica code irwinSums.m.

%F a(n+1) = A057168(a(n)). - _M. F. Hasler_, Aug 27 2014

%F a(n) = 2^A194882(n-1) + 2^A194883(n-1) + 2^A194884(n-1) + 2^A127324(n-1). - _Ridouane Oudra_, Sep 06 2020

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

%t Select[ Range[ 180 ], (Count[ IntegerDigits[ #, 2 ], 1 ]==4)& ] (* _Olivier Gérard_ *)

%o (Perl) $N = 4;

%o my $vector = 2 ** $N - 1;  # first key (15)

%o for (1..100) {

%o   print "$vector, ";

%o   my ($v, $d) = ($vector, 0);

%o   until ($v & 1 or !$v) { $d = ($d << 1)|1; $v >>= 1 }

%o   $vector += $d + 1 + (($v ^ ($v + 1)) >> 2);  # next key

%o } # _Ruud H.G. van Tol_, Mar 02 2014

%o (PARI) for(n=0,10^3,if(hammingweight(n)==4,print1(n,", "))); \\ _Joerg Arndt_, Mar 04 2014

%o (PARI) print1(t=15); for(i=2, 50, print1(", "t=A057168(t))) \\ _M. F. Hasler_, Aug 27 2014

%o (Python)

%o 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

%o (Rust)

%o pub const fn next_choice(value: usize) -> usize {

%o   // Passing a term will return the next number in the sequence

%o   let zeros = value.trailing_zeros();

%o   let ones = (value >> zeros).trailing_ones();

%o   value + (1 << zeros) + (1 << (ones - 1)) - 1

%o } // _Andrew Bennett_, Jan 07 2022

%Y Cf. A090706.

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

%Y Cf. A194882, A194883, A194884, A127324.

%K nonn,base,easy

%O 1,1

%A Al Black (gblack(AT)nol.net)

%E Extension by _Olivier Gérard_