A065722 -id:A065722 - cp's OEIS frontend

A065720 Primes whose binary representation is also the decimal representation of a prime.

3, 5, 23, 47, 89, 101, 149, 157, 163, 173, 179, 199, 229, 313, 331, 367, 379, 383, 443, 457, 523, 587, 631, 643, 647, 653, 659, 709, 883, 947, 997, 1009, 1091, 1097, 1163, 1259, 1277, 1283, 1289, 1321, 1483, 1601, 1669, 1693, 1709, 1753, 1877, 2063, 2069, 2099

Offset: 1

Patrick De Geest, Nov 15 2001

In general rebase notation (Marc LeBrun): p2 = (2) [p] (10).
Also: Primes in A036952. - M. F. Hasler, Dec 11 2012
See A089971 for the binary representation of these terms. - M. F. Hasler, Jan 05 2014

			1009{10} = 1111110001{2} is prime, and 1111110001{10} is also prime.
89 is in the sequence because it is a prime. Binary representation of 89 = 1011001, which is also a prime.

Harry J. Smith and K. D. Bajpai, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime
Carlos Rivera, Puzzle 280. 3893257, The Prime Puzzles & Problems Connection.

Cf. A065721, A065722, A065723, A065724, A065725, A065726, A065727, A065361.

Cf. A123266, A256621, A256622.

select(t -> isprime(t) and isprime(convert(t,binary)),[seq(2*i+1,i=1..1000)]); # Robert Israel, Jul 08 2014

Select[ Range[1900], PrimeQ[ # ] && PrimeQ[ FromDigits[ IntegerDigits[ #, 2]]] & ]
Select[ Prime@ Range@ 330, PrimeQ[ FromDigits[ IntegerDigits[#, 2]]] &] (* Robert G. Wilson v, Oct 09 2014 *)

isok(p) = isprime(p) && isprime(fromdigits(binary(p), 10)); \\ Michel Marcus, Mar 04 2022

from sympy import isprime
def ok(n): return isprime(n) and isprime(int(bin(n)[2:]))
print([k for k in range(2100) if ok(k)]) # Michael S. Branicky, Mar 04 2022

Equals A036952 intersect A000040. - M. F. Hasler, Dec 11 2012

a(48)-a(50) from K. D. Bajpai, Jul 04 2014

A065727 Primes p such that the decimal expansion of its base-9 conversion is also prime.

Original entry on oeis.org

2, 3, 5, 7, 37, 43, 61, 109, 127, 199, 271, 277, 379, 457, 487, 523, 541, 613, 619, 673, 727, 757, 883, 907, 919, 991, 997, 1033, 1117, 1249, 1447, 1483, 1531, 1549, 1567, 1627, 1693, 1699, 1747, 1753, 1987, 2053, 2161, 2221, 2287, 2341, 2347, 2437, 2473

Offset: 1

Patrick De Geest, Nov 15 2001

In general rebase notation (Marc LeBrun): p9 = (9) [p] (10).

			E.g., 997_10 = 1327_9 is prime, and so is 1327_10.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A065720, A065721, A065722, A065723, A065724, A065725, A065726, A065361.

Intersection of A000040 and A036964.

Select[ Range[2500], PrimeQ[ # ] && PrimeQ[ FromDigits[ IntegerDigits[ #, 9]]] & ]
NestList[NestWhile[NextPrime, #, ! PrimeQ[FromDigits[IntegerDigits[#2, 9]]] &, 2] &, 2, 48] (* Jan Mangaldan, Jul 01 2020 *)
Select[Prime[Range[400]],PrimeQ[FromDigits[IntegerDigits[#,9],10]]&] (* Harvey P. Dale, Sep 19 2021 *)

isok(p) = isprime(p) && isprime(fromdigits(digits(p, 9))); \\ Michel Marcus, Jul 02 2020

A281227 Primes whose binary reflected Gray code representation is also the decimal representation of a prime.

Original entry on oeis.org

2, 53, 233, 281, 397, 521, 613, 673, 733, 773, 797, 829, 1049, 1129, 1433, 1553, 1697, 1933, 2129, 2237, 2273, 2281, 2437, 2521, 2557, 2617, 2729, 2969, 3121, 3181, 3413, 3457, 3517, 3637, 3709, 3761, 3881, 4337, 4357, 4729, 4733, 4877, 4889, 5101, 5657, 5813, 5857, 6113, 6133

Offset: 1

Indranil Ghosh, Jan 18 2017

			521 is in the sequence because 521_10 = 1100001101_2 and both 521 and 1100001101 are prime numbers in base 10.

Indranil Ghosh, Table of n, a(n) for n = 1..481

Cf. A014550, A065720, A065721, A065722, A065723, A065724.

from sympy import isprime
def gray(n):
    return bin(n^(n//2))[2:]
i=1
j=1
while j<=481:
    if isprime(i)==True and isprime(int(gray(i)))==True:
        print(str(j)+" "+str(i))
        j+=1
    i+=1

A373033 a(0) = 5. For n >= 1, a(n) = a(n-1) converted to base 4 and interpreted in base 10.

Original entry on oeis.org

5, 11, 23, 113, 1301, 110111, 122320133, 13102213110011, 2332222120300201203323, 133212320111123130111021311111121323, 12122133133313032110200332320320202022333020121323230212223, 1323212003321221211122101013133003222123113122111221033300222032132012202011331212030120003001333

Offset: 0

Paolo Xausa, May 20 2024

a(n) is prime for n = 0, 1, 2, 3, 4. What is the next n for which a(n) is prime?

			5        -->        11        -->        23 ...
  base 10 to base 4    base 10 to base 4

Paolo Xausa, Table of n, a(n) for n = 0..15
Grant Sanderson, How They Fool Ya (live) | Math parody of Hallelujah, 3Blue1Brown YouTube video, 2023.

Cf. A007090, A023378, A065722.

NestList[FromDigits[IntegerDigits[#, 4]] &, 5, 11]

From Jianing Song, May 22 2024: (Start)
a(n+1) = A007090(a(n)).
As A007090(n+1) = A007090(n) + 1 if n is not congruent to 3 modulo 4, and all terms of A023378 are even, we have a(n) = A023378(n) + 1 by induction. (End)

A316479 a(n) is the smallest prime whose base-b expansion, read as a base-10 number, is a prime for every b in 2, 3, ..., n. (For n > 10, each base-b expansion for 10 < b <= n must contain no digit larger than 9.)

Original entry on oeis.org

3, 157, 157, 9241, 9241, 48404791, 18172964503, 50006393431, 50006393431, 181395559296673

Offset: 2

Jon E. Schoenfield, Jul 16 2018

a(2)=3, the smallest term in A065720, primes whose binary representation is also the decimal representation of a prime;
a(3)=157, the smallest integer in both A065720 and A065721, primes p whose base-3 expansion is also the decimal expansion of a prime;
similarly, a(4)=157 is the smallest integer in A065720, A065721, and A065722.
Is this sequence infinite?
a(12) > 10^16. - Giovanni Resta, Aug 01 2018

			a(2)=3 because 3 is prime, 3_10 = 11_2, and 11 is prime, and 3 is the smallest such number.
a(3)=157 because 157 is prime, 157_10 = 10011101_2, 157_10 = 12211_3, and 10011101 and 12211 are prime, and 157 is the smallest such number. a(4)=157 as well, since 157_10 = 2131_4 and 2131 is also prime.

Cf. A065720, A065721, A065722, A065723, A065724, A065725, A065726, A065727.

Cf. A084482, A236537.

a(8)-a(10) from Giovanni Resta, Jul 17 2018

a(11) from Giovanni Resta, Jul 24 2018

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A065720 Primes whose binary representation is also the decimal representation of a prime.

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A065727 Primes p such that the decimal expansion of its base-9 conversion is also prime.

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A281227 Primes whose binary reflected Gray code representation is also the decimal representation of a prime.

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A373033 a(0) = 5. For n >= 1, a(n) = a(n-1) converted to base 4 and interpreted in base 10.

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A316479 a(n) is the smallest prime whose base-b expansion, read as a base-10 number, is a prime for every b in 2, 3, ..., n. (For n > 10, each base-b expansion for 10 < b <= n must contain no digit larger than 9.)

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