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

0,5

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.

Links

Table of n, a(n) for n=0..19.

Formula

a(A385537(m)) = 1.

a(n) <= A277364(n).

Example

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).

Prog

(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

Crossrefs

Cf. A385537. Row sums of A378761.

Sequence in context: A393747 A300083 A062328 * A378199 A263918 A136234

Adjacent sequences: A385536 A385537 A385538 * A385540 A385541 A385542

Keyword

nonn,base,more

Author

Dmytro Inosov, Jul 02 2025

Status

approved