A376084 Number of cryptarithmically unique primes with n decimal digits.

0, 1, 0, 0, 0, 0, 2, 1, 3, 18, 105

Offset: 1

Author

Dmytro Inosov, Sep 09 2024

Keywords

• nonn
• base
• hard
• more

Comments

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.

It is conjectured that a(n) is asymptotic to A006879(n) as n->oo based on the combinatorial probability estimate under the assumption that asymptotically for large n, the fraction of primes among integers that share a given digit pattern would be the same as among all integers with n digits, given by p(n)=1/(n*ln10) according to the prime number theorem. Since the number of integers sharing the same digit pattern cannot exceed 9*9!, the probability for a randomly chosen prime of length n to be cryptarithmically unique >= (1-p(n))^(9*9!-1), which is asymptotic to 1 as n->oo.

The following terms 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(12)=24,

a(13)=668,

a(14)=1129,

a(15)=1306,

a(16)=4263,

a(17)=17320,

a(18)=6734,

a(19)=81794.

Further conjectured terms: a(20)=125975, a(21)=180471, a(22)=852579. - Michael S. Branicky, Oct 16 2024

Examples

			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.
		

Links

• Dmytro S. Inosov and Emil Vlasák, Cryptarithmically unique terms in integer sequences, arXiv:2410.21427 [math.NT], 2024.

Crossrefs

Cf. A374238 (cryptarithmically unique primes), A004022 (prime repunits), A358497, A376918.

Formula

a(n) <= A376918(n).

a(n) <= A006879(n).

lim_{n->oo} a(n)/A006879(n)=1 (conjectured).