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

Author

Cino Hilliard, Jan 18 2004

Keywords

• base
• nonn

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

Examples

			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.
		

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

Crossrefs

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

Programs

Mathematica

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

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

Extensions

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