A235461 Primes whose base-4 representation also is the base 2-representation of a prime.

5, 17, 257, 277, 337, 1093, 1109, 1297, 1361, 4357, 5189, 16453, 16657, 16661, 17489, 17669, 17681, 17749, 21521, 21569, 21589, 65537, 65557, 65617, 65809, 66821, 70657, 70981, 70997, 81937, 82241, 83221, 83269, 86017, 86357, 87317, 263429, 263489, 267541, 278549

Offset

1,1

Comments

This sequence is part of the two-dimensional array of sequences 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.

When the smaller base is b=2 such that only digits 0 and 1 are allowed, these are primes that are the sum of distinct powers of the larger base, here c=4, thus a subsequence of A077718 and therefore also of A000695, the Moser-de Bruijn sequence.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Example

5 = 11_4 and 11_2 = 3 are both prime, so 5 is a term.
17 = 101_4 and 101_2 = 5 are both prime, so 17 is a term.

Prog

(PARI) is(p, b=2, c=4)=vecmax(d=digits(p, c))<b&&isprime(vector(#d, i, b^(#d-i))*d~)&&isprime(p)
(Python)
from itertools import islice
from sympy import nextprime, isprime
def A235461_gen(): # generator of terms
    p = 1
    while (p:=nextprime(p)):
        if isprime(m:=int(bin(p)[2:], 4)):
            yield m
A235461_list = list(islice(A235461_gen(), 20)) # Chai Wah Wu, Aug 21 2023

Crossrefs

Cf. A090707 - A091924, A235462 - A235482. See the LINK for further cross-references.

Sequence in context: A191500 A089894 A077718 * A271660 A273948 A271657

Adjacent sequences: A235458 A235459 A235460 * A235462 A235463 A235464

Keyword

nonn,base

Author

M. F. Hasler, Jan 11 2014

Extensions

a(37)-a(40) from Robert Price, Nov 01 2023

Status

approved