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A065297 a(n+1) is the smallest number > a(n) such that the digits of a(n)^2 are all (with multiplicity) properly contained in the digits of a(n+1)^2, with a(0)=1.

%I A065297 #18 Jul 03 2025 18:21:53
%S A065297 1,4,13,36,113,487,1036,3214,10456,36786,100963,319656,1001964,
%T A065297 3165969,10001786,31626854,100013919,316256807,1000029656,3162322481,
%U A065297 10000115537
%N A065297 a(n+1) is the smallest number > a(n) such that the digits of a(n)^2 are all (with multiplicity) properly contained in the digits of a(n+1)^2, with a(0)=1.
%C A065297 Probably infinite and at least O(10^(n/2)). - _David W. Wilson_
%e A065297 13^2 = 169 and 36 is the next smallest number whose square (in this case 1296) properly contains the digits 1,6,9.
%o A065297 (Haskell)
%o A065297 import Data.List ((\\), sort)
%o A065297 a065297 n = a065297_list !! n
%o A065297 a065297_list = 1 : f 1 (drop 2 a000290_list) where
%o A065297    f x (q:qs) | null (xs \\ sq) && sort xs /= sort sq = y : f y qs
%o A065297               | otherwise                             = f x qs
%o A065297               where y = a000196 q; sq = show q; xs = show (x * x)
%o A065297 -- _Reinhard Zumkeller_, Nov 22 2012
%Y A065297 Cf. A050630, A050631, A048559, A050636, A065298, A014563, A066825.
%Y A065297 Cf. A000290, A000196.
%K A065297 base,nonn
%O A065297 0,2
%A A065297 _Floor van Lamoen_, Oct 29 2001
%E A065297 More terms from _Marc Paulhus_, Jan 29 2002
%E A065297 More terms from _David W. Wilson_ and _Marc Paulhus_, Feb 05 2002
%E A065297 a(19)-a(20) from _Sean A. Irvine_, Aug 26 2023