A235462 Primes whose base-5 representation also is the base-2 representation of a prime.

5, 31, 131, 151, 631, 3251, 3881, 19531, 78781, 78901, 81281, 81401, 81901, 82031, 94531, 97001, 97501, 390781, 394501, 406381, 469501, 471901, 472631, 484531, 1953901, 1956881, 1968751, 1969531, 1971901, 2031251, 2035151, 2046901, 2047651, 2050031, 2347001, 2360131

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 he 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=5, thus a subsequence of A077719.

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 = 10_5 and 10_2 = 2 are both prime, so 5 is a term.
31 = 111_5 and 111_2 = 7 are both prime, so 31 is a term.

Mathematica

b5b2Q[n_]:=Module[{idn5=IntegerDigits[n, 5]}, Max[idn5]<2 && PrimeQ[ FromDigits[ idn5, 2]]]; Select[Prime[Range[180000]], b5b2Q] (* Harvey P. Dale, Sep 21 2018 *)

Prog

(PARI) is(p, b=2, c=5)=vecmax(d=digits(p, c))<b&&isprime(vector(#d, i, b^(#d-i))*d~)&&isprime(p)
(Python)
from itertools import islice
from sympy import isprime, nextprime
def A235462_gen(): # generator of terms
    p = 1
    while (p:=nextprime(p)):
        if isprime(m:=int(bin(p)[2:], 5)):
            yield m
A235462_list = list(islice(A235462_gen(), 20)) # Chai Wah Wu, Aug 21 2023

Crossrefs

Sequence in context: A041303 A184446 A077719 * A005826 A293716 A294692

Adjacent sequences: A235459 A235460 A235461 * A235463 A235464 A235465

Keyword

nonn,base

Author

M. F. Hasler, Jan 11 2014

Status

approved