cp's OEIS Frontend

We call a k-tuple of binary vectors of length n complementary if for every position m (1 <= m <= n) the digit "1" occurs on that position in exactly one of the vectors. For example, {1010, 0100, 0001} is a triple (k=3) of complementary binary vectors of length n=4. The sum of complementary binary vectors of length n is always a repunit of the same length, A002275(n).

T(n,k) gives the number of distinct unordered k-tuples of complementary binary vectors of length n that have a common divisor > 1 as integers in base 10.

For k > n/2, at least one of the binary vectors must contain just a single "1" (with all other digits zero) and is, therefore, a power of 10 (A011557). Hence it cannot have nontrivial common divisors with the repunit A002275(n), which implies T(n,k) = 0. The requirement k <= n/2 acts to skip the corresponding trivial zero terms.

The partitions for k = 1 are trivial and consist of one element -- the repunit itself, which is its own greatest common divisor. Therefore, T(n,1) = 1 for n >= 2.

If T(n,k)=0 for some n and k, then T(n,m)=0 also for any m >= k. Indeed, if some m-tuple of binary vectors existed that is counted toward T(n,m), then an (m-1)-tuple obtained by summing any two of its vectors while leaving others unchanged would be counted toward T(n,m-1). By induction, this leads to T(n,k)>0, which is a contradiction.

Consequently, T(n,k) = 0 for all k > 1 whenever A378511(n) = 1. This holds, in particular, for all n in A004023 (indices of prime repunits).

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).

Two binary vectors of length n are mutually complementary if they are obtained from each other by inverting all the individual digits (1->0 and 0->1). The sum of any two mutually complementary binary vectors of length n is a repunit of the same length, A002275(n).

a(n) is the number of distinct unordered pairs of mutually complementary binary vectors of length n that have a common divisor > 1 as integers in base 10.

a(n) is the number of terms m in A007088 of length n that are not coprime with their 1's complement, A002275(n) - m, as integers in base 10.

Equivalently, m from A007088 is counted toward a(A055642(m)) iff GCD(m, A002275(A055642(m)) - m) > 1. Therefore a(n) is the base-10 analog of A378514.

a(n) is the sum of the first two columns, T(n,1) + T(n,2), in the triangle A378761.

For any n > 1, a(n) > 0 because the trivial partition A002275(n) = A002275(n) + 0 always counts toward a(n): GCD(A002275(n), 0) = A002275(n) > 1.

a(n) is nonmonotonic and tends to show minima for prime n, yet there are terms where n is prime yet a(n) > 1, for example a(13) = 131.

The sequence of indices n such that a(n) = 1 is a supersequence of A004023 (indices of prime repunits).

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