OFFSET
1,7
COMMENTS
Terms for n > 11 are conjectured based on the assumption that at these lengths A374238 does not contain terms with 4 or more distinct digits, which follows from the vanishing probability of such terms estimated with combinatorial arguments.
a(n) gives the number of n-digit primes p for which no other prime shares the same digit pattern, A358497(p).
a(n) is the count of terms in A374238 of length n.
a(n) shows anomalously small values for n divisible by 3 because certain digit patterns cannot result in primes based on divisibility rules: Whenever every digit occurs a number of times that is divisible by 3, the sum of digits is also divisible by 3, and therefore the number cannot be prime. For example, for n=12 all patterns consisting of 2 distinct digits A and B with the number of both A's and B's divisible by 3 (such as "AABABAAAABAA" and alike) cannot produce primes and therefore do not contribute to the total count. As a result, a(n) is not monotonic.
EXAMPLE
a(2)=1 because the only cryptarithmically unique prime (A374238) with 2 digits is 11. Indeed, any other 2-digit natural number with the same pattern "AA" is divisible by 11, whereas no 2-digit prime with the pattern "AB" of two nonequal digits is cryptarithmically unique because there are 20 primes that share the same pattern (all 2-digit primes except 11).
a(3)=0 because there are no cryptarithmically unique primes (A374238) with 3 digits.
a(7)=2 because there are exactly two cryptarithmically unique primes with 7 digits, which are 3333311 and 7771717.
CROSSREFS
KEYWORD
nonn,base,hard,more,new
AUTHOR
Dmytro Inosov, Sep 09 2024
EXTENSIONS
a(20) from Michael S. Branicky, Oct 03 2024
a(21) from Michael S. Branicky, Oct 07 2024
a(22) from Michael S. Branicky, Oct 16 2024
STATUS
approved