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.

%I A385539 #30 Jul 23 2025 16:14:25
%S A385539 1,0,1,1,4,1,26,1,175,365,1512,1,52611,274,426897,3072870,10670148,1,
%T A385539 879525398,1
%N 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.
%C A385539 Complementary binary vectors are as per A378761.
%C A385539 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.
%C A385539 For n <= 5, a(n) coincides with A378511(n).
%C A385539 Starting from n=2, a(n) gives the row sums of T(n,k) in A378761.
%C A385539 a(n) = 1 for all n in A004023 (indices of prime repunits).
%C A385539 a(n) = 1 iff n is a term in A385537.
%F A385539 a(A385537(m)) = 1.
%F A385539 a(n) <= A277364(n).
%e A385539 a(4) = A378511(4) = A378761(4,1) + A378761(4,2) = 4.
%e A385539 The only partition that counts toward A378761(4,1) is the trivial partition {1111} with only one part.
%e A385539 Among the possible pairs of nonzero binary vectors of length 4, exactly 3 are not coprime and therefore count toward A378761(4,2):
%e A385539   {1000,0111}: GCD(1000, 111) = 1;
%e A385539   {1001,0110}: GCD(1001, 110) = 11;
%e A385539   {1010,0101}: GCD(1010, 101) = 101;
%e A385539   {1011,0100}: GCD(1011, 100) = 1;
%e A385539   {1100,0011}: GCD(1100, 11) = 11;
%e A385539   {1101,0010}: GCD(1101, 10) = 1;
%e A385539   {1110,0001}: GCD(1110, 1) = 1.
%e A385539 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).
%o A385539 (Python)
%o A385539 from math import gcd
%o A385539 from sympy.utilities.iterables import multiset_partitions
%o A385539 def A385539(n):
%o A385539     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
%Y A385539 Cf. A385537. Row sums of A378761.
%K A385539 nonn,base,more
%O A385539 0,5
%A A385539 _Dmytro Inosov_, Jul 02 2025