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.

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

Author

Dmytro Inosov, Jul 03 2025

Keywords

• nonn
• base
• more

Comments

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).

Examples

			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.
		

Crossrefs

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

Programs

Mathematica

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}]

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

Formula

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