cp's OEIS Frontend

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.

%I A385579 #13 Jul 07 2025 19:34:22
%S A385579 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,
%T A385579 3191,3,2791,11,3,11,41,3,2028119,11,3,11,83,3,173,11,3,11,35121409,3,
%U A385579 239,11,3,11,107,3,41,11,3,11
%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.
%C A385579 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.
%C A385579 a(n)=1 iff n is a term in A385537 (indices of repunits coprime with all other binary vectors of the same length).
%F A385579 If a(n) > 1, A067063(n) <= a(n) <= A003020(n).
%e A385579 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.
%e A385579 a(4) = 11 because 11 is the smallest prime factor of 1111 which it shares, for example, with the binary vector 0011.
%t A385579 F[d_] := Min[Select[Table[Min[Transpose[FactorInteger[GCD[FromDigits[IntegerDigits[i,2]],(10^d-1)/9]]][[1]]], {i, 1, 2^d-2}], # > 1 &]];
%t A385579 Table[If[# < \[Infinity], #, 1] &[F[n]], {n, 0, 25}]
%o A385579 (PARI) 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
%Y A385579 Cf. A067063, A003020, A378761, A385537, A385539.
%K A385579 nonn,base,more
%O A385579 0,5
%A A385579 _Dmytro Inosov_, Jul 03 2025