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T(n,k) is defined as the number of distinct digit patterns (or digital types, as per A266946) of length n with k distinct digits such that all integers of that digital type share no common prime factor of a different digital type (as per A376918). Terms with k > n are omitted as trivial zeros.

To check whether a digit pattern of length n with k distinct digits A,B,... should be counted toward T(n,k), write that pattern as a linear combination of the form X1*A + X2*B + ..., where the pattern coefficients X1,X2,... consist of 0's and 1's (A007088), with 1's on positions of the corresponding digit in the pattern.

If GCD(X1,X2,...) has no prime divisors with a different digit pattern from the one we started from, the pattern is counted toward T(n,k). Otherwise, it is excluded.

For k = 10, the sum of all distinct digits A + B + ... + J = 45, which is divisible by 9. Hence, patterns in which all 10 distinct digits from A to J have the same number of occurrences in the pattern modulo 3 are identically divisible by 3 and should be excluded. This has an effect on T(n,10) whenever n takes the form 10m + 3q (with m >= 1 and q >= 0).

The digital types excluded in this way result in composites for any values of the distinct digits in the pattern, without the need to run primality tests on all numbers of that digital type individually.

The requirement for a divisor of a different digital type only affects terms with k=1, i.e. repdigits AA..AA, and acts to include that pattern iff the n-repunit is prime (n in A004023).

T(n,k) gives an upper bound for the number of contributions to A267013(n) with exactly k distinct digits.

Stirling numbers of the second kind S2(n,k) (A008277) give the total number of possible digital types of length n with k distinct digits and are therefore an upper bound for T(n,k).

T(n,1)=1 either when n=1 or when n is a term in A004023 (indices of prime repunits); otherwise T(n,1)=0 because all the repdigits A*(10^n-1)/9 are simultaneously divisibly by any proper divisor of the repunit (10^n-1)/9.

T(n,2) is nonmonotonic because a larger number of digit patterns is excluded whenever n has a large number of nontrivial divisors (A070824), resulting in anomalously low values, for example, for n=12 or n=18. This is a consequence of divisibility rules that are formulated for prime divisors of 10^r-1 or 10^r+1 (where r divides n) in terms of the sum or alternating sum of r-digit blocks, respectively [see S. Shirali, First Steps in Number Theory: A Primer on Divisibility, Universities Press, 2019, pp. 42-49].

The sequence is related to A266991.

Sequence {A164864(n) - a(n)}_(n>=1) begins 0,0,1,4,1,26,1,175,365,1516,...

One can explain, why, for example, a(4)=11, instead of A164864(4)=15. There exist exactly 4 types of 4-digit numbers, which cannot be prime. In A266946 these types are: 1001, 1010, 1100, 1111. Indeed, numbers abba,aabb,aaaa are divisible by 11; a number abab is divisible by 101.

In other cases of n-digit types we should verify the divisibility of numbers of types in A266946 at least by primes of the form 11,101,... Besides, a digital type 1...1 exists only for n in A004023, i.e., for only 9 values of n from the first 270343. This simplifies the calculations.

a(n) <= A376918(n) with equality for n <= 9, but thereafter some digital types which pass the divisibility rules of A376918 don't in fact occur among the primes (see A377727). - Dmytro Inosov, Nov 05 2024

Based on the conjectured terms in A377727, the next three terms can be conjectured: a(14)=190402538; a(15)=1378294708; a(16)=10437142874. - Dmytro Inosov, Jan 07 2025

The digit pattern for any natural number n is uniquely defined by the canonical form A358497(n), which enumerates digits according to their position of first occurrence. Each prime in this sequence has a unique digit pattern in the sense that no other prime has the same pattern.

Prime repunits (A004022) are a subsequence, as they are the sole primes with a single distinct digit.

A cryptarithm (alphametic) expresses a digit pattern in letters, where each distinct letter is to map to a distinct digit.

If a cryptarithmetic problem calls for a prime number, then the primes in this sequence are unique solutions, so we call these primes cryptarithmically unique.

The smallest term with 3 distinct digits is 1151135331533311.

The number of terms of length n is given by A376084(n).

Digit patterns (or digital types) are as per A266946.

The divisibility rules are per A376918 and they act to exclude patterns which always result in composite numbers, just due to the pattern.

There are A376918(n) remaining patterns but not all of them actually contain primes, and a(n) is how many of them do not, so that a(n) = A376918(n) - A267013(n).

We call these digital types primonumerophobic and a(n) is the number of these of length n.

It is conjectured that the next terms are a(14)=362, a(15)=363, a(16)=1448. This is based on the calculated number of primonumerophobic digit patterns with only 2 or 3 distinct digits and the vanishingly small combinatorial probability for the existence of additional primonumerophobic digit patterns of this length with 4 or more distinct digits.

a(n) gives the number of distinct digit patterns (or digital types, as per A266946) such that all integers of that digital type share a common prime factor of a different digital type.

The number of remaining digit patterns not counted toward a(n) is given by A376918(n).

A particular digit pattern of length n is counted toward a(n) if it is not counted toward A376918(n).

All digital types counted toward a(n) result in composites for any values of the distinct digits in the pattern, without the need to run primality tests on all numbers of that digital type individually.

The requirement for a divisor of a different digital type only affects reprigits of the form AA..AA and acts to exclude that pattern iff the n-repunit is prime (n in A004023).

A164864(n) gives the total number of possible digit patterns of length n and is therefore an upper bound for a(n).

a(n) is nonmonotonic and takes on small values for prime n and large values for n with a large number of nontrivial divisors (A070824). This is a consequence of divisibility rules that are formulated for prime divisors of 10^r-1 or 10^r+1 (where r divides n) in terms of the sum or alternating sum of r-digit blocks, respectively [see S. Shirali, First Steps in Number Theory: A Primer on Divisibility, Universities Press, 2019, pp. 42-49].

a(n) coincides with row sums of T(n,k) in A378761 for n = 3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 17, 18. - Dmytro Inosov, Dec 23 2024

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