cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

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

Views

Author

Bernard Schott, Oct 12 2022

Keywords

Comments

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).

Crossrefs

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.

Programs

  • Maple
    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
  • Mathematica
    Select[Range[1, 500, 2], DigitCount[#, 2, 0] == 2 &] (* Amiram Eldar, Oct 12 2022 *)
  • PARI
    isok(k) = (k%2) && (#binary(k) == hammingweight(k)+2); \\ Michel Marcus, Oct 13 2022
  • PARI
    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
  • Python
    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
  • Python
    # 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
  • Python
    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

Formula

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

Extensions

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

A360574 Binary expansions of odd numbers with three zeros in their binary expansion.

Original entry on oeis.org

10001, 100011, 100101, 101001, 110001, 1000111, 1001011, 1001101, 1010011, 1010101, 1011001, 1100011, 1100101, 1101001, 1110001, 10001111, 10010111, 10011011, 10011101, 10100111, 10101011, 10101101, 10110011, 10110101, 10111001, 11000111, 11001011, 11001101, 11010011, 11010101, 11011001, 11100011
Offset: 1

Views

Author

Bernard Schott, Feb 18 2023

Keywords

Comments

For m >= 5, there are A000292(m-4) terms with m digits.

Examples

			1010101 has three digits 0 and is the binary expansion of the odd integer 85, so 1010101 is a term.

Crossrefs

Cf. A000292, A007088, A360573.
Similar, but with k zeros in their binary expansion: A000042 (k=0), A190619 (k=1), A357774 (k=2).

Programs

  • Mathematica
    FromDigits[IntegerDigits[#, 2]] & /@ Select[Range[1, 250, 2], DigitCount[#, 2, 0] == 3 &] (* Amiram Eldar, Feb 18 2023 *)
  • Python
    from itertools import count, islice
from sympy.utilities.iterables import multiset_permutations
def A360574_gen(): # generator of terms
    yield from (int('1'+''.join(d)+'1') for l in count(0) for d in  multiset_permutations('000'+'1'*l))
A360574_list = list(islice(A360574_gen(),30)) # Chai Wah Wu, Feb 18 2023

Formula

a(n) = A007088(A360573(n)).
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.