cp's OEIS Frontend

A294660 Least nonnegative integer not occurring earlier whose square has no digit in common with the square of the previous term, a(0) = 0.

%I A294660 #23 Aug 11 2024 14:41:34
%S A294660 0,1,2,3,4,5,6,7,9,8,10,15,12,16,20,11,22,13,18,14,28,19,17,21,23,26,
%T A294660 29,24,30,25,33,58,27,34,47,38,45,31,48,41,50,37,52,44,65,40,57,76,32,
%U A294660 63,35,60,39,62,36,88,46,67,51,183,75,43,55,42,53,56,70,61,64,85,59,77,69,73,78,89
%N A294660 Least nonnegative integer not occurring earlier whose square has no digit in common with the square of the previous term, a(0) = 0.
%C A294660 This is not a permutation of the nonnegative integers, since numbers whose square has all digits '1' through '9' (cf. A294661, e.g., 11826 with 11826^2 = 139854276) can never appear - and these numbers have asymptotic density 1.
%C A294660 Will all integers whose square does not have all of the digits 1-9, eventually appear? Or might the sequence be finite? Since a(n)^2 has no digits in common with a(n-1)^2, it is sufficient for a(n+1) to exist, to find a number whose square has a subset of the digits of a(n-1)^2. Is this always possible? This problem sometimes has only "sporadic k-digital solutions", see, e.g., A058430, A030175, ... and the link to De Geest's page.
%H A294660 M. F. Hasler, <a href="/A294660/b294660.txt">Table of n, a(n) for n = 0..4829</a>
%H A294660 P. De Geest, <a href="https://www.worldofnumbers.com/threedigits.htm">Squares containing at most three distinct digits, Index entries for related sequences</a>
%e A294660 Since a(7)^2 = 7^2 = 49, the subsequent term cannot be 8, since 8^2 = 64 has the digit 4 in common with 49. Therefore, a(8) = 9, with 9^2 = 81 having no common digit with 49.
%e A294660 a(1201) = 1037. So the square of the next term must not have any of the digits in {0, 1, 3, 5, 6, 7, 9}, only 2, 4, 8 are allowed. The least such number that has not been used before is a(1202) = 210912978, with a(1202)^2 = 210912978^2 = 44484284288828484. - _Alois P. Heinz_, Nov 09 2017
%o A294660 (PARI) {u=a=0; for(n=0, 99, print1(a", "); u+=1<<a; D=Set(if(a, digits(a^2))); for(k=1, oo, bittest(u, k)&&next; #setintersect(D, Set(digits(k^2)))&&next; a=k; break)); a}
%o A294660 (PARI) {u=[a=0]; for(n=0, 99, print1(a", "); D=Set(if(a, digits(a^2))); for(k=u[1]+1, oo, setsearch(u, k)&&next; #setintersect(D, Set(digits(k^2)))&&next; u=setunion(u,[a=k]); break); while(#u>1&&u[2]==u[1]+1,u=u[^1])); a}
%Y A294660 Cf. A030287 (strictly increasing), A067581 (do not take squares).
%K A294660 nonn,base,look
%O A294660 0,3
%A A294660 _M. F. Hasler_, Nov 08 2017