A190620 -id:A190620 - cp's OEIS frontend

A030130 Binary expansion contains a single 0.

0, 2, 5, 6, 11, 13, 14, 23, 27, 29, 30, 47, 55, 59, 61, 62, 95, 111, 119, 123, 125, 126, 191, 223, 239, 247, 251, 253, 254, 383, 447, 479, 495, 503, 507, 509, 510, 767, 895, 959, 991, 1007, 1015, 1019, 1021, 1022, 1535, 1791, 1919, 1983, 2015, 2031, 2039

Offset: 1

Toby Donaldson (tjdonald(AT)uwaterloo.ca)

From Reinhard Zumkeller, Aug 29 2009: (Start)
A023416(a(n)) = 1;
apart from the initial term the sequence can be seen as a triangle read by rows, see A164874;
A055010 and A086224 are subsequences, see also A000918 and A036563. (End)
Zero and numbers of form 2^m-2^k-1, 2 <= m, 0 <= k <= m-2. - Zak Seidov, Aug 06 2010

			23 is OK because it is '10111' in base 2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000918, A023416, A036563, A055010, A086224, A140977, A160502, A164874, A190620.

long int element (long int i) { return (pow(2,g(i))-1-pow(2,(pow(2*g(i)-1,2)-1-8*i)/8));} long int g(long int m) {if (m==0) return(1); return ((sqrt(8*m-7)+3)/2);}

a030130 n = a030130_list !! (n-1)
a030130_list = filter ((== 1) . a023416) [0..]
-- Reinhard Zumkeller, Mar 31 2015, Dec 07 2012

[0] cat [k:k in [0..2050]| Multiplicity(Intseq(k,2),0) eq 1]; // Marius A. Burtea, Feb 06 2020

Sort[Flatten[{{0}, Table[2^n - 2^m - 1, {n, 2, 50}, {m, 0, n - 2}]}]] (* Zak Seidov, Aug 06 2010 *)
Select[Range[0,2100],DigitCount[#,2,0]==1&] (* Harvey P. Dale, Dec 19 2021 *)

print1("0, ");for(k=1,2039,my(v=digits(k,2));if(vecsum(v)==#v-1,print1(k,", "))) \\ Hugo Pfoertner, Feb 06 2020

from math import isqrt, comb
def A030130(n): return (1<<(a:=(isqrt(n-1<<3)+1>>1)+1))-(1<Chai Wah Wu, Dec 19 2024

a(n) = 2^(g(n))-1-2^(((2*g(n)-1)^2-1-8*n)/8) with g(n)=int((sqrt(8*n-7)+3)/2) for all n>0 and g(0)=1. - Ulrich Schimke (ulrschimke(AT)aol.com)
a(n+1) = A140977(a(n)) for any n > 1. - Rémy Sigrist, Feb 06 2020
Sum_{n>=2} 1/a(n) = A160502. - Amiram Eldar, Oct 06 2020
a(n) = (A190620(n-1)-1)/2. - Chai Wah Wu, Dec 19 2024

More terms from Erich Friedman

Offset fixed by Reinhard Zumkeller, Aug 24 2009

A357773 Odd numbers with two zeros in their binary expansion.

Original entry on oeis.org

9, 19, 21, 25, 39, 43, 45, 51, 53, 57, 79, 87, 91, 93, 103, 107, 109, 115, 117, 121, 159, 175, 183, 187, 189, 207, 215, 219, 221, 231, 235, 237, 243, 245, 249, 319, 351, 367, 375, 379, 381, 415, 431, 439, 443, 445, 463, 471, 475, 477, 487, 491, 493, 499, 501

Offset: 1

Bernard Schott, Oct 12 2022

A048490 \ {1} is a subsequence, since for m >= 1, A048490(m) = 8*2^m - 7 has 11..11001 with m starting 1 for binary expansion.
A153894 \ {4} is a subsequence, since for m >= 1, A153894(m) = 5*2^m - 1 has 10011..11 with m trailing 1 for binary expansion.
A220236 is a subsequence, since for m >= 1, A220236(m) = 2^(2*m + 2) - 2^(m + 1) - 2^m - 1 has 11..110011..11 with m starting 1 and m trailing 1 for binary expansion.
For k > 2, there are (k-1)*(k-2)/2 terms between 2^k and 2^(k+1), or equivalently (k-1)*(k-2)/2 terms with k+1 bits.
Binary expansion of a(n) is A357774(n).
{4*a(n), n>0} form a subsequence of A353654 (numbers with two trailing 0 bits and two other 0 bits).

Cf. A018900, A005408, A023416, A353654, A357774.
Odd numbers with k zeros in their binary expansion: A000225 (k=0), A190620 (k=1).
Subsequences: A048490 \ {1}, A153894 \ {4}, A220236.

seq(seq(seq(2^n-1-2^i-2^j,j=i-1..1,-1),i=n-2..1,-1),n=4..10); # Robert Israel, Oct 13 2022

Select[Range[1, 500, 2], DigitCount[#, 2, 0] == 2 &] (* Amiram Eldar, Oct 12 2022 *)

isok(k) = (k%2) && (#binary(k) == hammingweight(k)+2); \\ Michel Marcus, Oct 13 2022

list(lim)=my(v=List()); for(n=4,logint(lim\=1,2)+1, my(N=2^n-1); forstep(a=n-2,2,-1, my(A=N-1<lim, break(2)); listput(v,t)))); Vec(v) \\ Charles R Greathouse IV, Oct 21 2022

def a(n):
    m = 0
    while m*(m+1)*(m+2)//6 <= n: m += 1
    m -= 1 # m = A056556(n-1)
    k, r, j = m + 4, n - m*(m+1)*(m+2)//6, 0
    while r >= 0: r -= (m+1-j); j += 1
    j += 1
    return 2**k - 2**(k-j) - 2**(-r) - 1
print([a(n) for n in range(60)]) # Michael S. Branicky, Oct 12 2022

# faster version for generating initial segment of sequence
from itertools import combinations, count, islice
def agen():
    for d in count(4):
        b, c = 2**d - 1, 2**(d-1)
        for i, j in combinations(range(1, d-1), 2):
            yield b - (c >> i) - (c >> j)
print(list(islice(agen(), 60))) # Michael S. Branicky, Oct 13 2022

from math import comb, isqrt
from sympy import integer_nthroot
def A357773(n):
    a = (m:=integer_nthroot(6*n, 3)[0])+(n>comb(m+2,3))+3
    b = isqrt((j:=comb(a-1,3)-n+1)<<3)+3>>1
    c = j-comb((r:=isqrt(w:=j<<1))+(w>r*(r+1)),2)
    return (1<Chai Wah Wu, Dec 17 2024

A023416(a(n)) = 2.

a((n-1)*(n-2)*(n-3)/6 - (i-1)*(i-2)/2 - (j-1)) = 2^n - 2^i - 2^j - 1 for 1 <= j < i <= n-2. - Robert Israel, Oct 13 2022

a(11) and beyond from Michael S. Branicky, Oct 12 2022

A095078 Primes with a single 0 bit in their binary expansion.

Original entry on oeis.org

2, 5, 11, 13, 23, 29, 47, 59, 61, 191, 223, 239, 251, 383, 479, 503, 509, 991, 1019, 1021, 2039, 3583, 3967, 4079, 4091, 4093, 6143, 15359, 16127, 16319, 16381, 63487, 65407, 65519, 129023, 131063, 245759, 253951, 261631, 261887, 262079

Offset: 1

Antti Karttunen, Jun 01 2004

Except for the first value 2, the sequence gives the primes of the form 2^k -2^j -1 with 0 < j < k-1. If j=k-1 we obtain the Mersenne primes. - Pierre CAMI, May 19 2005

{2} UNION (A000040 INTERSECT A190620). - Chai Wah Wu, Dec 19 2024

T. D. Noe, Table of n, a(n) for n=1..1000
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 and A030130. Cf. A095058, A190620.

Select[Prime[Range[23000]],DigitCount[#,2,0]==1&] (* Harvey P. Dale, Nov 28 2019 *)

forprime(p=2,262079,v=binary(p);s=0;for(k=1,#v,s+=v[k]);if(#v-s==1,print1(p,", "))) \\ Washington Bomfim, Jan 13 2011

A353654 Numbers whose binary expansion has the same number of trailing 0 bits as other 0 bits.

Original entry on oeis.org

1, 3, 7, 10, 15, 22, 26, 31, 36, 46, 54, 58, 63, 76, 84, 94, 100, 110, 118, 122, 127, 136, 156, 172, 180, 190, 204, 212, 222, 228, 238, 246, 250, 255, 280, 296, 316, 328, 348, 364, 372, 382, 392, 412, 428, 436, 446, 460, 468, 478, 484, 494, 502, 506, 511, 528, 568

Offset: 1

Mikhail Kurkov, Jul 15 2022

Numbers k such that A007814(k) = A086784(k).
To reproduce the sequence through itself, use the following rule: if binary 1xyz is a term then so are 11xyz and 10xyz0 (except for 1 alone where 100 is not a term).
The number of terms with bit length k is equal to Fibonacci(k-1) for k > 1.
Conjecture: 2*A247648(n-1) + 1 with rewrite 1 -> 1, 01 -> 0 applied to binary expansion is the same as a(n) without trailing 0 bits in binary.
Odd terms are positive Mersenne numbers (A000225), because there is no 0 in their binary expansion. - Bernard Schott, Oct 12 2022

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000045, A000120, A000523, A007814, A010056, A025480, A030109, A054429, A060142, A072649, A084471, A086784, A091892, A132665, A133512, A200650, A213911, A232559, A280514, A343152, A348366.
Cf. A356385 (first differences).
Subsequences with same number k of trailing 0 bits and other 0 bits: A000225 (k=0), 2*A190620 (k=1), 4*A357773 (k=2), 8*A360573 (k=3).

N:= 10: # for terms <= 2^N
S:= {1};
for d from 1 to N do
  for k from 0 to d/2-1 do
    B:= combinat:-choose([$k+1..d-2],k);
    S:= S union convert(map(proc(t) local s; 2^d - 2^k - add(2^(s),s=t) end proc,B),set);
od od:
sort(convert(S,list)); # Robert Israel, Sep 21 2023

Join[{1}, Select[Range[2, 600], IntegerExponent[#, 2] == Floor[Log2[# - 1]] - DigitCount[# - 1, 2, 1] &]] (* Amiram Eldar, Jul 16 2022 *)

isok(k) = if (k==1, 1, (logint(k-1, 2)-hammingweight(k-1) == valuation(k, 2))); \\ Michel Marcus, Jul 16 2022

from itertools import islice, count
def A353654_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:(m:=(~n & n-1).bit_length()) == bin(n>>m)[2:].count('0'),count(max(startvalue,1)))
A353654_list = list(islice(A353654_gen(),30)) # Chai Wah Wu, Oct 14 2022

a(n) = a(n-1) + A356385(n-1) for n > 1 with a(1) = 1.
Conjectured formulas: (Start)
a(n) = 2^g(n-1)*(h(n-1) + 2^A000523(h(n-1))*(2 - g(n-1))) for n > 2 with a(1) = 1, a(2) = 3 where f(n) = n - A130312(n), g(n) = [n > 2*f(n)] and where h(n) = a(f(n) + 1).
a(n) = 1 + 2^r(n-1) + Sum_{k=1..r(n-1)} (1 - g(s(n-1, k)))*2^(r(n-1) - k) for n > 1 with a(1) = 1 where r(n) = A072649(n) and where s(n, k) = f(s(n, k-1)) for n > 0, k > 1 with s(n, 1) = n.
a(n) = 2*(2 + Sum_{k=1..n-2} 2^(A213911(A280514(k)-1) + 1)) - 2^A200650(n) for n > 1 with a(1) = 1.
A025480(a(n)-1) = A348366(A343152(n-1)) for n > 1.
a(A000045(n)) = 2^(n-1) - 1 for n > 1. (End)

A360573 Odd numbers with exactly three zeros in their binary expansion.

Original entry on oeis.org

17, 35, 37, 41, 49, 71, 75, 77, 83, 85, 89, 99, 101, 105, 113, 143, 151, 155, 157, 167, 171, 173, 179, 181, 185, 199, 203, 205, 211, 213, 217, 227, 229, 233, 241, 287, 303, 311, 315, 317, 335, 343, 347, 349, 359, 363, 365, 371, 373, 377, 399, 407, 411, 413

Offset: 1

Bernard Schott, Feb 12 2023

If m is a term then 2*m+1 is another term, since if M is the binary expansion of m, then M.1 where . stands for concatenation is the binary expansion of 2*m+1.
A052996 \ {1,3,8} is a subsequence, since for m >= 3, A052996(m) = 9*2^(m-2) - 1 has 100011..11 with m-2 trailing 1 for binary expansion.
A171389 \ {20} is a subsequence, since for m >= 1, A171389(m) = 21*2^m - 1 has 1010011..11 with m trailing 1 for binary expansion.
A198276 \ {18} is a subsequence, since for m >= 1, A198276(m) = 19*2^m - 1 has 1001011..11 with m trailing 1 for binary expansion.
Binary expansion of a(n) is A360574(n).
{8*a(n), n>0} form a subsequence of A353654 (numbers with three trailing 0 bits and three other 0 bits).
Numbers of the form 2^(a+1) - 2^b - 2^c - 2^d - 1 where a > b > c > d > 0. - Robert Israel, Feb 13 2023

			35_10 = 100011_2, so 35 is a term.

Cf. A005408, A023416, A056557, A333516, A353654, A360010, A360574.
Subsequences: A052996 \ {1,3,8}, A171389 \ {20}, A198276 \ {18}.
Odd numbers with k zeros in their binary expansion: A000225 (k=0), A190620 (k=1), A357773 (k=2), this sequence (k=3).

q:= n-> n::odd and add(1-i, i=Bits[Split](n))=3:
select(q, [$1..575])[];  # Alois P. Heinz, Feb 12 2023
# Alternative:
[seq(seq(seq(seq(2^(a+1) - 2^b - 2^c - 2^d - 1, d = c-1..1,-1), c=b-1..2,-1),b=a-1..3,-1),a=4..12)]; # Robert Israel, Feb 13 2023

Select[Range[1, 500, 2], DigitCount[#, 2, 0] == 3 &] (* Amiram Eldar, Feb 12 2023 *)

isok(m) = (m%2) && #select(x->(x==0), binary(m)) == 3; \\ Michel Marcus, Feb 13 2023

def ok(n): return n&1 and bin(n)[2:].count("0") == 3
print([k for k in range(414) if ok(k)]) # Michael S. Branicky, Feb 12 2023

from itertools import count, islice
from sympy.utilities.iterables import multiset_permutations
def A360573_gen(): # generator of terms
    yield from (int('1'+''.join(d)+'1',2) for l in count(0) for d in  multiset_permutations('000'+'1'*l))
A360573_list = list(islice(A360573_gen(),54)) # Chai Wah Wu, Feb 18 2023

from itertools import combinations, count, islice
def agen(): yield from ((1<Michael S. Branicky, Feb 18 2023

from math import comb, isqrt
from sympy import integer_nthroot
def A056557(n): return (k:=isqrt(r:=n+1-comb((m:=integer_nthroot(6*(n+1), 3)[0])-(nA333516(n): return (r:=n-1-comb((m:=integer_nthroot(6*n, 3)[0])+(n>comb(m+2, 3))+1, 3))-comb((k:=isqrt(m:=r+1<<1))+(m>k*(k+1)), 2)+1
def A360010(n): return (m:=integer_nthroot(6*n, 3)[0])+(n>comb(m+2, 3))
def A360573(n):
    a = (a2:=integer_nthroot(24*n, 4)[0])+(n>comb(a2+2, 4))+3
    j = comb(a-1,4)-n
    b, c, d = A360010(j+1)+2, A056557(j)+2, A333516(j+1)
    return (1<Chai Wah Wu, Dec 18 2024

A023416(a(n)) = 3.

Let a = floor((24n)^(1/4))+4 if n>binomial(floor((24n)^(1/4))+2,4) and a = floor((24n)^(1/4))+3 otherwise. Let j = binomial(a-1,4)-n. Then a(n) = 2^a-1-2^(A360010(j+1)+2)-2^(A056557(j)+2)-2^(A333516(j+1)). - Chai Wah Wu, Dec 18 2024

A190619 Binary expansions of odd numbers with a single zero in their binary expansion.

Original entry on oeis.org

101, 1011, 1101, 10111, 11011, 11101, 101111, 110111, 111011, 111101, 1011111, 1101111, 1110111, 1111011, 1111101, 10111111, 11011111, 11101111, 11110111, 11111011, 11111101, 101111111, 110111111, 111011111, 111101111, 111110111, 111111011, 111111101, 1011111111, 1101111111, 1110111111, 1111011111, 1111101111, 1111110111, 1111111011, 1111111101

Offset: 1

N. J. A. Sloane, May 14 2011

a(n) = A007088(A190620(n)). [Reinhard Zumkeller, May 14 2011]

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

a190619 n = a190619_list !! (n-1)
a190619_list = map read $ f 2 1 :: [Integer] where
  f m k
    | k < m - 1 = ((take k ones) ++ "0" ++ (take (m-k) ones)) : f m (k+1)
    | otherwise = ((take k ones) ++ "01") : f (m + 1) 1
  ones = repeat '1'
-- Reinhard Zumkeller, May 15 2011

f:=k->(10^k-1)/9; for n from 3 to 12 do for i from n-2 by -1 to 1 do j:=f(n)-10^i; lprint(j); od; od;

from itertools import count, islice
def agen():
    for d in count(3):
        b = (10**d - 1)//9
        for i in range(2, d):
            yield b - 10**(d-i)
print(list(islice(agen(), 50))) # Michael S. Branicky, Oct 13 2022

from math import isqrt, comb
def A190619(n): return (10**((a:=(isqrt(n<<3)+1>>1)+1)+1)-1)//9-10**(comb(a,2)-n+1) # Chai Wah Wu, Dec 18 2024

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A030130 Binary expansion contains a single 0.

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A357773 Odd numbers with two zeros in their binary expansion.

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A095078 Primes with a single 0 bit in their binary expansion.

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A353654 Numbers whose binary expansion has the same number of trailing 0 bits as other 0 bits.

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A360573 Odd numbers with exactly three zeros in their binary expansion.

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A190619 Binary expansions of odd numbers with a single zero in their binary expansion.

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