A350577 Prime numbers in A036991.

3, 5, 7, 11, 13, 19, 23, 29, 31, 43, 47, 53, 59, 61, 71, 79, 83, 103, 107, 109, 127, 151, 157, 167, 173, 179, 181, 191, 199, 211, 223, 239, 251, 271, 283, 307, 311, 317, 331, 347, 349, 359, 367, 373, 379, 383, 431, 439, 443, 461, 463, 467, 479, 487, 491, 499

Offset: 1

Author

Gennady Eremin, Jan 07 2022

Keywords

• nonn
• base

Comments

This sequence includes A000668.

Conjecture: The sequence is infinite. For example, in the first million primes (see A000040) 304208 numbers are terms of A036991.

Links

• Gennady Eremin, Table of n, a(n) for n = 1..20000
• Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024. See pp. 10, 14.

Crossrefs

Subsequence of A000040, A005408, A036991, A042987, A045395, A095070, A095074, A216285.

Cf. A000668.

Programs

Maple

q:= proc(n) local l, t, i; l:= Bits[Split](n); t:=0;
      for i to nops(l) do t:= t-1+2*l[i];
        if t<0 then return false fi
      od: true
    end:
select(isprime and q, [$2..500])[];  # Alois P. Heinz, Jan 07 2022

Mathematica

q[n_] := PrimeQ[n] && AllTrue[Accumulate[(-1)^Reverse[IntegerDigits[n, 2]]], # <= 0 &]; Select[Range[500], q] (* Amiram Eldar, Jan 07 2022 *)

Python

from sympy import isprime
def ok(n):
    if n == 0: return True
    count = {"0": 0, "1": 0}
    for bit in bin(n)[:1:-1]:
        count[bit] += 1
        if count["0"] > count["1"]: return False
    return isprime(n)
print([k for k in range(3, 500, 2) if ok(k)]) # Michael S. Branicky, Jan 07 2022

Formula

Intersection of A000040 and A036991.