cp's OEIS Frontend

Complementary binary vectors are as per A378761.

a(n) gives the total number of distinct unordered tuples of complementary binary vectors of length n (including those with leading zeros) that have a common divisor > 1 as integers in base 10. Since any such tuple sums up to the repunit A002275(n), it corresponds to an integer partition of the repunit.

For n <= 5, a(n) coincides with A378511(n).

Starting from n=2, a(n) gives the row sums of T(n,k) in A378761.

a(n) = 1 for all n in A004023 (indices of prime repunits).

a(n) = 1 iff n is a term in A385537.

a(n) is the smallest prime factor that divides both the decimal repunit (10^n-1)/9 and at least one other smaller decimal number consisting of only 0's and 1's.

a(n)=1 iff n is a term in A385537 (indices of repunits coprime with all other binary vectors of the same length).

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

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).

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) counts binary numbers (A007088) of length n that are not coprime with their bitwise inverse as integers in base 2.

Equivalently, m from A007088 is counted toward a(A055642(m)) iff GCD(m, A002275(A055642(m)) - m) > 1, assuming base 2 in the calculation of GCD. Therefore a(n) is the base-2 analog of A378511.

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) = 1 iff n is a Mersenne exponent (A000043). Indeed, if a partition 2^n-1 = k + m exists with GCD(k, m) = q > 1, then 2^n-1 itself is divisible by q. Whenever 2^n-1 is a Mersenne prime (A000668), this is only possible for q = 2^n-1, therefore the only such partition is the trivial one, {2^n-1, 0}. The inverse is also true. If n is not a Mersenne exponent, 2^n-1 has a nontrivial divisor q, and the partition 2^n-1 = q + (2^n-1-q) counts toward a(n) because GCD(q, 2^n-1-q) = q > 1. Therefore, a(n) > 1.

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

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].

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 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).