A385537 -id:A385537 - cp's OEIS frontend

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

Dmytro Inosov, Jul 02 2025

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(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).

Cf. A385537. Row sums of A378761.

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

a(A385537(m)) = 1.

a(n) <= A277364(n).

A385579 Smallest prime factor that the repunit(n) = (10^n-1)/9 shares with at least one other binary vector of the same length in base 10, or 1 if they are coprime.

Original entry on oeis.org

1, 1, 1, 1, 11, 1, 3, 1, 11, 3, 11, 1, 3, 53, 11, 3, 11, 1, 3, 1, 11, 3, 11, 1, 3, 41, 11, 3, 11, 3191, 3, 2791, 11, 3, 11, 41, 3, 2028119, 11, 3, 11, 83, 3, 173, 11, 3, 11, 35121409, 3, 239, 11, 3, 11, 107, 3, 41, 11, 3, 11

Offset: 0

Dmytro Inosov, Jul 03 2025

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).

			a(3) = 1 because 111 = 3*37 is coprime with all other nonzero binary vectors of length 3, which are 001, 010, 011, 100, 101, 110. None of them is divisible by 3 or 37.
a(4) = 11 because 11 is the smallest prime factor of 1111 which it shares, for example, with the binary vector 0011.

Cf. A067063, A003020, A378761, A385537, A385539.

F[d_] := Min[Select[Table[Min[Transpose[FactorInteger[GCD[FromDigits[IntegerDigits[i,2]],(10^d-1)/9]]][[1]]], {i, 1, 2^d-2}], # > 1 &]];
Table[If[# < \[Infinity], #, 1] &[F[n]], {n, 0, 25}]

a(n) = my(x=(10^n-1)/9, m=oo, b=0, z); for (i=1, 2^n-2, my(y=fromdigits(binary(i))); if ((z=gcd(y, x)) != 1, b=1; m = min(m, vecmin(factor(z)[,1])); ); ); if (b, m, 1); \\ Michel Marcus, Jul 03 2025

If a(n) > 1, A067063(n) <= a(n) <= A003020(n).

cp's OEIS Frontend

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.

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