A361173 Numbers k such that, in base 4, the greatest prime less than 4^k and the least prime greater than 4^k have no common digit.

1, 4, 28, 83, 1816

Offset

1,2

Comments

In base 4 all consecutive primes with no common digit are of this form, except for 2 and 3.

It is unknown whether this sequence is infinite.

Base 2 and base 3 have no such primes.

Links

Table of n, a(n) for n=1..5.

Mathematics StackExchange, Are there an infinity of disjoint consecutive primes?

Example

k=4 is a term: the consecutive primes are 251 and 257. In base 4 their representations are 3323 and 10001, which have no common digit.

Mathematica

Select[Range[100], ! IntersectingQ @@ IntegerDigits[NextPrime[4^#, {-1, 1}], 4] &] (* Amiram Eldar, Mar 03 2023 *)

Prog

(PARI) isok(k) = #setintersect(Set(digits(precprime(4^k), 4)), Set(digits(nextprime(4^k), 4))) == 0; \\ Michel Marcus, Mar 03 2023
(Python)
from sympy.ntheory import digits, nextprime, prevprime
def ok(n):
    p, q = prevprime(4**n), nextprime(4**n)
    return set(digits(p, 4)[1:]) & set(digits(q, 4)[1:]) == set()
print([k for k in range(1, 99) if ok(k)]) # Michael S. Branicky, Mar 03 2023

Crossrefs

Cf. A104082, A104089, A068802, A156981.

Sequence in context: A085001 A153784 A030117 * A005634 A183485 A183437

Adjacent sequences: A361170 A361171 A361172 * A361174 A361175 A361176

Keyword

nonn,base,more

Author

Lewis Baxter, Mar 02 2023

Status

approved