A256622 -id:A256622 - 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

A123266 Primes p such that the decimal expansion of p remains prime under two iterations of base-10 to base-2 conversions.

Original entry on oeis.org

5, 1097, 2237, 2689, 3541, 12979, 13477, 22367, 22783, 27701, 28499, 33521, 33613, 43093, 51839, 55487, 57383, 65423, 69931, 70201, 71429, 74209, 80599, 82267, 82889, 83591, 95009, 99079, 99881, 105929, 122201, 123923, 125261

Offset: 1

Martin Raab, Oct 09 2006

More precisely, "... remains prime under two iterations of base-10 to base-2 conversions, but not three iterations."

.

			5 is a term because 5_10 = 101_2 and 101_10 = 1100101_2 and both 101 and 1100101 are prime in base 10.

Sebastian Petzelberger, Table of n, a(n) for n = 1..10000.
Carlos Rivera, Puzzle 280. 3893257, The Prime Puzzles & Problems Connection.

Cf. A065720, A007088, A256621, A256622.

okQ[n_] := And @@ PrimeQ[Rest[NestList[FromDigits[IntegerDigits[#, 2]] &, n, 2]]]; Select[Prime[Range[20000]], okQ] (* Harvey P. Dale, Jan 14 2011 *)

A007088(n)=fromdigits(binary(n),10)
is(n)=isprime(n) && isprime(n=A007088(n)) && isprime(A007088(n)) \\ Charles R Greathouse IV, Apr 08 2015

A256621 Primes p such that the decimal expansion of p remains prime under three iterations of base-10 to base-2 conversion.

Original entry on oeis.org

3893257, 9023533, 11005327, 11659009, 18747809, 21855233, 25183007, 34074379, 54298687, 58562951, 60496981, 89891273, 94277683, 96602887, 102276859, 115555927, 117578429, 122191543, 125115709, 132837283, 138169991, 139442753, 168852077, 183879649, 184904831

Offset: 1

Sebastian Petzelberger, Apr 06 2015

			The 3 iterations: 3893257 --> 1110110110100000001001 --> ... --> ... are prime.

Sebastian Petzelberger, Table of n, a(n) for n = 1..10000
Carlos Rivera, Puzzle 280. 3893257, The Prime Puzzles and Problems Connection.

Cf. A065720, A123266, A256622.

isok(n, nb=3) = {for (k=1, nb, b = binary(n); d = eval(subst(Pol(b), x, 10)); if (! isprime(d), return (0)); n = d;); return (1);} \\ Michel Marcus, Apr 08 2015

A361312 Smallest prime p such that the decimal expansion of p remains prime through exactly n iterations of base-10 to base-2 conversion (A007088).

Original entry on oeis.org

2, 3, 5, 3893257, 9632552297

Offset: 0

Ya-Ping Lu, Mar 08 2023

Prime numbers that remain primes after 1, 2, 3, and 4 iterations are A065720, A123266, A256621 and A256622, respectively.

			a(0) = 2 because prime number 2 in base 2 is 10, and 10 in base 10 is not a prime.
a(1) = 3 because 3 = 11_2 and 11_10 is a prime. In the second iteration, however, 11_10 = 1011_2 and 1011_10 is not a prime.
a(2) = 5 because 5 = 101_2 and 101_10 = 1100101_2. Both 101 and 1100101 are primes in base 10. In the third iteration, 1100101_10 = 100001100100101000101_2 and 100001100100101000101_10 is not a prime.

Carlos Rivera, Puzzle 280. 3893257, The Prime Puzzles & Problems Connection.

Cf. A007088, A036952, A065720, A123266, A256621, A256622.

from sympy import isprime, nextprime
p = 1; mx = 5; I = [*range(mx)]; R = [*range(mx)]
while I:
    p = nextprime(p); ct = 0; q = p
    while isprime(int(bin(q)[2:])): ct += 1; q = int(bin(q)[2:])
    if ct in I: R[ct] = p; I.remove(ct)
print(*R, sep = ", ")

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

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A123266 Primes p such that the decimal expansion of p remains prime under two iterations of base-10 to base-2 conversions.

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A256621 Primes p such that the decimal expansion of p remains prime under three iterations of base-10 to base-2 conversion.

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A361312 Smallest prime p such that the decimal expansion of p remains prime through exactly n iterations of base-10 to base-2 conversion (A007088).

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