A216285 -id:A216285 - cp's OEIS frontend

A235478 Primes whose base-2 representation also is the base-8 representation of a prime.

7, 11, 13, 29, 37, 43, 47, 53, 61, 67, 71, 73, 107, 139, 149, 199, 211, 227, 263, 293, 307, 311, 317, 331, 347, 383, 389, 421, 461, 467, 541, 593, 601, 619, 641, 643, 739, 811, 863, 907, 937, 1061, 1069, 1093, 1117, 1163, 1223, 1283, 1301, 1319, 1321, 1409, 1433, 1439, 1489, 1499, 1523, 1559, 1619, 1697, 1811, 1861, 1879

Offset: 1

M. F. Hasler, Jan 12 2014

This sequence is part of a two-dimensional array of sequences, given in the LINK, based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.
For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.
Appears to be a subsequence of A050150, A062090 and A216285.

			11 = 1011_2 and 1011_8 = 521 are both prime, so 11 is a term.

Robert Price, Table of n, a(n) for n = 1..2265
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235465 ⊂ A077722, A235266, A152079, A235475 - A235479, A065720 ⊂ A036952, A065721 - A065727, A089971 ⊂ A020449, A089981, A090707 - A091924, A235394, A235395, A235461 - A235482. See the LINK for further cross-references.

Select[Prime[Range[300]],PrimeQ[FromDigits[IntegerDigits[#,2],8]]&] (* Harvey P. Dale, Sep 25 2015 *)

is(p,b=8)=isprime(vector(#d=binary(p),i,b^(#d-i))*d~)&&isprime(p)

A350577 Prime numbers in A036991.

Original entry on oeis.org

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

Gennady Eremin, Jan 07 2022

This sequence includes A000668.

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

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.

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

Cf. A000668.

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

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

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

Intersection of A000040 and A036991.

cp's OEIS Frontend

A235478 Primes whose base-2 representation also is the base-8 representation of a prime.

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