A053738 -id:A053738 - cp's OEIS frontend

A346310 Positions of words in A076478 such that #0's - #1's is even.

3, 4, 5, 6, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106

Offset: 1

Clark Kimberling, Aug 28 2021

			The first fourteen words w(n) are 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, 100, 101, 110, 111, so that a(1) = 3.

Cf. A007931, A076478, A346309 (complement), A053738.

Mathematica
```
(See A076478.)
```

a(n) = A053738(n+1) - 1, conjectured.

A380788 Numbers with a prime number of binary digits.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106

Offset: 1

Michael S. Branicky, Feb 03 2025

			4 is a term since its binary representation has 3 bits, a prime.
64 is a term since its binary representation has 7 bits, a prime.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A272441, A053738.

Select[Range[200], PrimeQ[BitLength[#]] &] (* Paolo Xausa, Feb 03 2025 *)

from sympy import isprime
def ok(n): return isprime(n.bit_length())
print([k for k in range(150) if ok(k)])

# faster for initial segment of sequence
from itertools import islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    d = 2
    while True:
        yield from (i for i in range(2**(d-1), 2**d))
        d = nextprime(d)
print(list(islice(agen(), 65)))

from sympy import primerange
def A380788(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(min(x,(1<Chai Wah Wu, Feb 03 2025

A079112 Numbers in binary representation with odd length.

Original entry on oeis.org

0, 1, 100, 101, 110, 111, 10000, 10001, 10010, 10011, 10100, 10101, 10110, 10111, 11000, 11001, 11010, 11011, 11100, 11101, 11110, 11111, 1000000, 1000001, 1000010, 1000011, 1000100, 1000101, 1000110, 1000111, 1001000, 1001001

Offset: 0

Reinhard Zumkeller, Dec 25 2002

a(n) = A007088(A053738(n)).

Join[{0},Select[FromDigits/@Tuples[{0,1},7],OddQ[IntegerLength[#]]&]] (* Harvey P. Dale, Sep 04 2021 *)

A333470 Lexicographically earliest sequence of distinct positive terms such that a(n) is the number of commas that a(n) has to step over (to the right) to be met by an odd term. This odd term might not be the closest odd term to a(n).

Original entry on oeis.org

1, 3, 2, 4, 5, 6, 7, 9, 8, 11, 10, 13, 12, 15, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 32, 35, 34, 37, 36, 39, 38, 41, 40, 43, 42, 45, 44, 47, 46, 49, 48, 51, 50, 53, 52, 55, 54, 57, 56, 59, 58, 61, 60, 63, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Eric Angelini and Carole Dubois, Mar 23 2020

			a(1) = 1 steps over 1 comma to be met by the odd term 3;
a(2) = 3 steps over 3 commas to be met by the odd term 5;
a(3) = 2 steps over 2 commas to be met by the same odd term 5;
a(4) = 4 steps over 4 commas to be met by the odd term 9 (the odd term 5 is closer, but this is not the point);
a(5) = 5 steps over 5 commas to be met by the odd term 11 (again, the odd terms 7 and 9 are closer, but we don't care); etc.

Dominic McCarty, Table of n, a(n) for n = 1..10000

def a(n): return n if len(bin(n))%2 else n-1 if n%2 else n+1 # Dominic McCarty, Mar 12 2025

a(n) = n for n in A053738. Otherwise, a(n) = n+1 for even n and a(n) = n-1 for odd n. - Dominic McCarty, Mar 12 2025

cp's OEIS Frontend

A346310 Positions of words in A076478 such that #0's - #1's is even.

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A380788 Numbers with a prime number of binary digits.

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A079112 Numbers in binary representation with odd length.

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A333470 Lexicographically earliest sequence of distinct positive terms such that a(n) is the number of commas that a(n) has to step over (to the right) to be met by an odd term. This odd term might not be the closest odd term to a(n).

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