A077712 a(1) = 1, a(n) = the smallest squarefree number > a(n-1) which contains all the digits of a(n-1).

1, 10, 101, 110, 1001, 1010, 10001, 10010, 100001, 100010, 1000001, 1000010, 10000001, 10000010, 100000001, 100000010, 1000000001, 1000000010, 10000000001, 10000000010, 100000000010, 1000000000001, 10000000000001

Offset: 1

Author

Amarnath Murthy, Nov 19 2002

Keywords

• base
• nonn

Comments

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

Crossrefs

Cf. A086981, A086982.

Subsequence of A005117.

Extensions

More terms from Ray Chandler, Aug 02 2003

Offset corrected by Mohammed Yaseen, Aug 16 2023