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a(n) gives the number of distinct digit patterns (or digital types, as per A266946) such that all integers of that digital type share no common prime factor of a different digital type.

The number of remaining digit patterns not counted toward a(n) is given by A378199(n).

To check whether a digit pattern of length n with distinct digits A,B,... should be counted toward a(n), 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 a(n). Otherwise, it is excluded.

For n of the form 10m + 3q (with m >= 1 and q >= 0), check in addition if the pattern contains all 10 distinct digits whose number of occurrences taken modulo 3 is the same for all digits from A to J. Since A + B + ... + J = 45, which is divisible by 9, such patterns are not counted toward a(n) and should be excluded.

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 reprigits of the form AA..AA and acts to include that pattern iff the n-repunit is prime (n in A004023).

a(n) gives the upper bound for the number of distinct digital types of n-digit primes A267013(n). The two sequences are distinct, since some digit patterns such as "AAABBCABCCCAACCB" happen to contain no primes accidentally, without having a common divisor. We call such patterns primonumerophobic. Sequence A377727 = {a(n)-A267013(n)}_(n>=1) gives the number of primonumerophobic digit patterns of length n.

a(n) coincides with A267013(n) for n<10 because the shortest primonumerophobic digit patterns "AAABBBAAAB", "AABABBBBBA", and "ABAAAAABBB" have length 10.

a(n) represents row sums of T(n,k) in A378154 -- an array of contributions to a(n) with exactly k<=10 distinct decimal digits.

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

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].

k is a term iff A378511(k) <= 1.

k is a term iff A385539(k) <= 1.

a(n) contains all indices of prime repunits A004023 as a subsequence.

If A378199(k) <= 1, then k is a term in this sequence, however the inverse is not true. The smallest counterexample is k = 19.

a(11) >= 59. - Michael S. Branicky, Jul 03 2025