cp's OEIS Frontend

This sequence is the union of the collection of sequences formed from the nonzero terms of A086981 * A005408, the odd numbers. First occurrence of consecutive integers in sequence is 406,407,408.

From Robert Israel, Feb 13 2017: (Start)

Numbers n such that gcd(n, 10^n + 1) > 1 or n = k*m where k is odd and 2*m is the order of 10 modulo a member of A045616. [Corrected by Jianing Song, Nov 01 2024]

If n is in the sequence, then so is k*n for any odd k. (End)

Numbers of the form k*ord(10,p^2)/2, where k is an odd number and p is a prime such that ord(10,p) is even. Here ord(a,m) is the multiplicative order of a modulo m. Note that if p is not in A045616, then ord(10,p^2) = p*ord(10,p). - Jianing Song, Nov 01 2024

Conjecture: Terms contain only two types of digits, i.e., 0 and 1.

Beginning with a(3), sequence follows a regular pattern: 10^2+1, 10^2+10, 10^3+1, 10^3+10, etc. until at a(21) the pattern is disrupted by 10^11+1, which is not squarefree (see A086982). 10^12+10 is also absent from the sequence since it is also not squarefree. The pattern resumes after this disruption until the next occurrence of 10^k+1 which is not squarefree, k=21, 33, 39, 55, ... The conjecture that the sequence is composed of terms containing only the digits 0 and 1 is certainly true up to 10^406+1 where both it and 10^407+1 are not squarefree. Indeed beginning with a(3) the terms contain exactly two 1 digits and the rest 0's up to this point. The term following 10^406+10 will introduce a third nonzero digit, perhaps a 1, but the pattern of the sequence changes dramatically at this point. - Ray Chandler, Aug 02 2003

Term following a(739)=10^406+10 is a(740)=10^407+11 so the conjecture is still in play. - Ray Chandler, Aug 05 2003