A359840 Numbers k that are the representation of primes in base 4 and in base 5.

2, 3, 23, 131, 133, 221, 1211, 1231, 2023, 2111, 2113, 2311, 3013, 3211, 3233, 3323, 10031, 10033, 10121, 12011, 12121, 13223, 13331, 20131, 20203, 22111, 23233, 31313, 32033, 32303, 33133, 33331, 100123, 100211, 100231, 101003, 101333, 103333, 110021, 111211

Offset

1,1

Comments

For a(1) = 2, 2_4 = 2_5 = 2_10 and for a(2) = 3, 3_4 = 3_5 = 3_10; otherwise, these two primes are distinct for n >= 3 (example).

The corresponding sequences of primes are A235474 (for base 4) and A235615 (for base 5).

Links

Table of n, a(n) for n=1..40.

Formula

a(n) = A007090(A235474(n)); a(n) = A007091(A235615(n)).

Example

a(3) = 23 because 23_4 = 11_10 = A235474(3) and 23_5 = 13_10 = A235615(3) are primes.
a(9) = 2023 because 2023_4 = 139_10 = A235474(9) and 2023_5 = 263_10 = A235615(9) are primes.

Mathematica

q[n_, b_] := Max[d = IntegerDigits[n]] < b && PrimeQ[FromDigits[d, b]]; Select[Range[200000], q[#, 4] && q[#, 5] &] (* Amiram Eldar, Jan 15 2023 *)

Prog

(Python)
from sympy import isprime
def ok(n): return max(s:=str(n)) < '4' and isprime(int(s, 4)) and isprime(int(s, 5))
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jan 15 2023
(Python)
from sympy import isprime
from itertools import count, islice, product
def agen(): yield from (int(s) for d in count(1) for f in "123" for r in product("0123", repeat=d-1) if isprime(int(s:=f+"".join(r), 4)) and isprime(int(s, 5)))
print(list(islice(agen(), 40))) # Michael S. Branicky, Jan 15 2023

Crossrefs

Intersection of A004678 and A004679.

Cf. A007090, A007091, A235474, A235615, A340290.

Sequence in context: A093504 A370279 A009180 * A041459 A126702 A260126

Adjacent sequences: A359837 A359838 A359839 * A359841 A359842 A359843

Keyword

nonn,base

Author

Bernard Schott, Jan 15 2023

Extensions

More terms from Amiram Eldar, Jan 15 2023

Status

approved