A385539 Total number of distinct partitions of the repunit A002275(n) into an arbitrary number of complementary binary vectors having a common divisor > 1 in base 10.

1, 0, 1, 1, 4, 1, 26, 1, 175, 365, 1512, 1, 52611, 274, 426897, 3072870, 10670148, 1, 879525398, 1

Offset: 0

Author

Dmytro Inosov, Jul 02 2025

Keywords

• nonn
• base
• more

Comments

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.

Examples

			a(4) = A378511(4) = A378761(4,1) + A378761(4,2) = 4.
The only partition that counts toward A378761(4,1) is the trivial partition {1111} with only one part.
Among the possible pairs of nonzero binary vectors of length 4, exactly 3 are not coprime and therefore count toward A378761(4,2):
  {1000,0111}: GCD(1000, 111) = 1;
  {1001,0110}: GCD(1001, 110) = 11;
  {1010,0101}: GCD(1010, 101) = 101;
  {1011,0100}: GCD(1011, 100) = 1;
  {1100,0011}: GCD(1100, 11) = 11;
  {1101,0010}: GCD(1101, 10) = 1;
  {1110,0001}: GCD(1110, 1) = 1.
Longer tuples cannot count toward a(4) because for any of them at least one of its binary vectors must contain just a single "1" (with all other digits zero). It is, therefore, a power of 10 (A011557) and cannot have nontrivial common divisors with the repunit A002275(n).
		

Crossrefs

Cf. A385537. Row sums of A378761.

Programs

Python

from math import gcd
from sympy.utilities.iterables import multiset_partitions
def A385539(n):
    return sum(1 for p in multiset_partitions([10**k for k in range(n)]) if gcd(*(sum(t) for t in p))!=1) # Pontus von Brömssen, Jul 16 2025

Formula

a(A385537(m)) = 1.

a(n) <= A277364(n).