A217050 Number of primes of n base-3 digits such that each of the 3 digits (0, 1, 2) appears exactly a prime number of times.

0, 0, 0, 0, 0, 0, 42, 164, 349, 1400, 3414, 7816, 30166, 30315, 241385, 93803, 1621424, 256109, 9504808, 2382005, 58680809, 5173582, 442095498, 27083394, 3763874067, 227444355, 20058458866, 500609319

Offset

1,7

Comments

We have a(n) = 0 for n < 7 because no prime represented in base 3 can have exactly two ones (or any larger even number).

Links

Table of n, a(n) for n=1..28.

Example

The smallest number in the class concerned is 773, which is represented as 1001122 in base 3. There are 41 other 7-digit numbers with the same digits in base 3 that are also prime.

Mathematica

Table[cnt = 0; Do[If[PrimeQ[i], d = IntegerDigits[i, 3]; c = Transpose[Tally[d]][[2]]; If[PrimeQ[c] == {True, True, True}, cnt++]], {i, 3^(n - 1), 3^n - 1}]; cnt, {n, 13}] (* T. D. Noe, Sep 27 2012 *)

Prog

(Python)
from sympy import primerange
from gmpy2 import is_prime, mpz
from sympy.utilities.iterables import multiset_permutations as mp
def a(n): return sum(1 for i in primerange(2, n-3)
                       for j in primerange(3, n-i-1)
                       if is_prime(k:=n-i-j)
                       for f in "12"
                       for m in mp(('0'*i+'1'*j+'2'*k).replace(f, '', 1))
                       if is_prime(mpz(f+"".join(m), 3)))
print([a(n) for n in range(1, 11)]) # Michael S. Branicky, Jul 21 2025

Crossrefs

Sequence in context: A159226 A160329 A010022 * A134385 A232329 A115130

Adjacent sequences: A217047 A217048 A217049 * A217051 A217052 A217053

Keyword

nonn,base,more

Author

James G. Merickel, Sep 25 2012

Extensions

a(20)-a(22) from Chai Wah Wu, Nov 12 2015

a(23)-a(26) from Michael S. Branicky, Jul 21 2025

a(27)-a(28) from Michael S. Branicky, Aug 18 2025

Status

approved