A089971 Primes whose decimal representation also represents a prime in base 2.

11, 101, 10111, 101111, 1011001, 1100101, 10010101, 10011101, 10100011, 10101101, 10110011, 11000111, 11100101, 100111001, 101001011, 101101111, 101111011, 101111111, 110111011, 111001001, 1000001011, 1001001011, 1001110111, 1010000011, 1010000111, 1010001101

Offset

1,1

Comments

See A065720 for the primes given by these terms considered as numbers written in base 2, i.e., the sequence with the definition "working in the opposite sense". - M. F. Hasler, Jan 05 2014

A subsequence of A020449. - M. F. Hasler, Jan 11 2014

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Alejandro J. Becerra Jr., Python code for computing terms of A089971, A089981, A090707-A090710, A235394-A235395.

Alejandro J. Becerra Jr., Table of n, a(n) for n = 1..42012

Example

a(1)=11 is a prime and its decimal representation is also a valid base-2 representation (because all digits are < 2), and 11_2 = 3_10 is again a prime.

Mathematica

Select[ FromDigits@# & /@ IntegerDigits[ Prime@ Range@ 270, 2], PrimeQ] (* Robert G. Wilson v, Jan 05 2014 *)

Prog

(PARI) is_A089971(p)=vecmax(d=digits(p))<2&&isprime(vector(#d, i, 2^(#d-i))*d~)&&isprime(p) \\ "d" is implicitly declared local. Putting isprime(p) to the end improves performance when the function is applied to primes only or to very large numbers. - M. F. Hasler, Jan 05 2014
(PARI) fixBase(n, oldBase, newBase)=my(d=digits(n, oldBase), t=newBase-1); for(i=1, #d, if(d[i]>t, for(j=i, #d, d[j]=t); break)); fromdigits(d, newBase)
list(lim)=my(v=List(), t); forprime(p=2, fixBase(lim\1, 10, 2), if(isprime(t=fromdigits(digits(p, 2), 10)), listput(v, t))); Vec(v) \\ Charles R Greathouse IV, Nov 07 2016
(Python)
from sympy import isprime, primerange
def aupto(limit):
    alst = []
    for p in primerange(2, limit+1):
        t = int(bin(p)[2:])
        if isprime(t): alst.append(t)
    return alst
print(aupto(2**11)) # Michael S. Branicky, Aug 19 2021

Crossrefs

Cf. A031974, A089981, A090707, A090708, A090709, A090710, A235394, A235395, A000040 and references therein.

Sequence in context: A080439 A098153 A020449 * A082620 A199304 A156668

Adjacent sequences: A089968 A089969 A089970 * A089972 A089973 A089974

Keyword

base,nonn

Author

Cino Hilliard, Jan 18 2004

Extensions

Definition and example reworded, offset corrected, and cross-references added by M. F. Hasler, Jan 05 2014

Status

approved