This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A385515 #18 Jul 08 2025 07:51:26
%S A385515 2,3,4,7,8,9,22,33,44,77,88,333,444,3333,33333,44444,88888,333333,
%T A385515 3333333,33333333,333333333,3333333333,33333333333,333333333333,
%U A385515 3333333333333,33333333333333,333333333333333,3333333333333333,33333333333333333,333333333333333333
%N A385515 Repdigit numbers whose square does not contain the repeated digit.
%C A385515 For n >= 18, all terms are of the form 33...3; that is, elements of A002277.
%C A385515 A002277(m) is a term, for m > 0. Proof: 3 is in a(n) because 3^2 = 9. If 33...3 is composed of k 3's, with k > 1, it is satisfied that 33...3^2 = 11...1088...89; i.e., (k - 1) 1's followed by a 0, then (k - 1) 8's and a 9, so that 3 is not among the digits of its square.
%C A385515 Let's see that there are no other terms of the form 33...3 besides 2, 4, 7, 8, 9, 22, 44, 77, 88, 444, 44444, 88888. In this sequence there are no repdigits of the form 11...1, 55...5, 66...6, since their squares end in 1, 5 and 6 respectively. On the other hand, 9 is the only number of the form 999...9, since if it has 2 or more 9's its square starts with 9. Suppose that dd...d contains 6 or more digits. We already saw that the cases d = 1, 5, 6 and 9 are discarded. Let us analyze what happens for d = 2, 4, 7 and 8:
%C A385515 For d = 2, we have that 22...2^2 == 284 (mod 10^3).
%C A385515 For d = 4, we have that 44...4^2 == 469136 (mod 10^6).
%C A385515 For d = 7, we have that 77...7^2 == 729 (mod 10^3).
%C A385515 For d = 8, we have that 88...8^2 == 876544 (mod 10^6).
%C A385515 Thus, we conclude that a(n) only consists of digits 3 for n >= 18. And, in fact, a(n) consists of (n - 12) 3's.
%F A385515 a(n) = A002277(n - 12), for n >= 18.
%e A385515 22 is a term since 22^2 = 484 does not contain the digit 2.
%t A385515 Select[Union@ Flatten@ Table[k (10^n - 1)/9, {k, 0, 9}, {n, 18}] ,ContainsNone[IntegerDigits[#^2],IntegerDigits[#]]&] (* _James C. McMahon_, Jul 07 2025 *)
%Y A385515 Cf. A002277, A010785, A029783.
%Y A385515 Intersection of A010785 and A029783.
%K A385515 nonn,base
%O A385515 1,1
%A A385515 _Gonzalo MartÃnez_, Jul 01 2025