A380786 Numbers with a prime number of bits, prime number of ones, and prime number of zeros in their binary representation.

17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 65, 66, 68, 72, 79, 80, 87, 91, 93, 94, 96, 103, 107, 109, 110, 115, 117, 118, 121, 122, 124, 4097, 4098, 4100, 4104, 4112, 4128, 4160, 4224, 4352, 4608, 5119, 5120, 5631, 5887, 6015, 6079, 6111, 6127, 6135, 6139, 6141, 6142, 6144, 6655, 6911

Offset

1,1

Comments

Each term has either two zeros or two ones in its binary representation. And so the total number of bits of each term is the larger prime of a twin prime pair.

Links

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

Example

a(1) = 17 = 10001_2. Number of bits is 5, number of ones is 2, number of zeros is 3. {2,3,5} are all primes.

Mathematica

Select[Range[2^13], AllTrue[{#1, #2, #1 - #2} & @@ {IntegerLength[#, 2], DigitCount[#, 2, 1]}, PrimeQ] & ] (* Michael De Vlieger, Feb 03 2025 *)

Prog

(Python)
from sympy import isprime
def ok(n): return isprime(L:=n.bit_length()) and isprime(O:=n.bit_count()) and isprime(L-O)
print([k for k in range(7160) if ok(k)]) # Michael S. Branicky, Feb 03 2025
(Python)
from sympy import isprime, nextprime, sieve
from itertools import combinations, count, islice
def agen(): # generator of terms
    p = 5
    while True:
        passed = set()
        if isprime(p-2):
            for q in [2, p-2]:
                for locs in combinations(range(1, p), q-1):
                    w = ["1"] + ["0"]*(p-1)
                    for i in locs: w[i] = "1"
                    passed.add(int("".join(w), 2))
        yield from sorted(passed)
        p = nextprime(p)
        print(len(passed), p)
print(list(islice(agen(), 61))) # Michael S. Branicky, Feb 03 2025
(PARI) isok(k) = my(h=hammingweight(k), b=#binary(k)); isprime(h) && isprime(b) && isprime(b-h); \\ Michel Marcus, Feb 07 2025

Crossrefs

Intersection of A052294, A144754, and A380788.

Subsequence of primes gives A272478.

Subsequence of A343258.

Cf. A006512 (bitlengths of terms).

Sequence in context: A217543 A296757 A297287 * A018822 A304488 A043594

Adjacent sequences: A380783 A380784 A380785 * A380787 A380788 A380789

Keyword

nonn,base

Author

Marc Morgenegg, Feb 03 2025

Status

approved