A385537 Indices k such that the repunit (10^k-1)/9 is coprime with any other nonzero binary vector of the same length in base 10.

0, 1, 2, 3, 5, 7, 11, 17, 19, 23

Offset

1,3

Comments

k is a term iff A378511(k) <= 1.

k is a term iff A385539(k) <= 1.

a(n) contains all indices of prime repunits A004023 as a subsequence.

If A378199(k) <= 1, then k is a term in this sequence, however the inverse is not true. The smallest counterexample is k = 19.

a(11) >= 59. - Michael S. Branicky, Jul 03 2025

Links

Table of n, a(n) for n=1..10.

Formula

A385579(a(n)) = 1.

Example

0 is a term because A002275(0) = 0, which is coprime with the only other binary vector of the same length, which is 1.
1 is a term because A002275(1) = 1, there are no other nonzero binary vectors of length 1, and any statement about the elements of an empty set is true.
2 is a term because 11 is a repunit prime.
3 is a term 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.
Counterexample: 4 is not a term because the repunit 1111 is not coprime with 1100. They are both divisible by 11.

Prog

(PARI) isok(k) = my(x=(10^k-1)/9); for (i=1, 2^k-2, if (gcd(fromdigits(binary(i)), x) != 1, return(0)); ); return(1); \\ Michel Marcus, Jul 03 2025

Crossrefs

Cf. A378199, A378511, A378761, A385539, A385579.

Supersequence of A004023.

Sequence in context: A050298 A094751 A280409 * A336370 A396301 A396303

Adjacent sequences: A385534 A385535 A385536 * A385538 A385539 A385540

Keyword

nonn,base,more

Author

Dmytro Inosov, Jul 02 2025

Status

approved