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A265155 Integers which are unique starting points for the algorithm described in A090566.

%I A265155 #19 Aug 08 2025 04:43:20
%S A265155 1,2,4,8,10,11,14,15,16,17,18,19,21,22,23
%N A265155 Integers which are unique starting points for the algorithm described in A090566.
%C A265155 Consider the family of integer sequences generated from a starting value b(1) and the rule that each subsequent term is the smallest number greater than the previous term such that the concatenation of the two is a square. Then using
%C A265155   b(1) = a(1) =  1 yields {1, 6, 25, 281, 961, ...}    (A090566),
%C A265155   b(1) = a(2) =  2 yields {2, 5, 29, 241, 1809, ...}   (A265147),
%C A265155   b(1) = a(3) =  4 yields {4, 9, 61, 504, 4516, ...}   (A265148),
%C A265155   b(1) = a(4) =  8 yields {8, 41, 209, 764, 5225, ...} (A265149),
%C A265155   b(1) = a(5) = 10 yields {10, 24, 336, 400, 689, ...} (A265150),
%C A265155   b(1) = a(6) = 11 yields {11, 56, 169, 744, 769, ...} (A265151),
%C A265155   ...
%e A265155 The complement of {a(n)} is {3, 5, 6, 7, 9, 12, 13, 20, ...}; using any of these values as b(1) yields a sequence that quickly merges into one of the sequences obtained using a value from {a(n)} as b(1):
%e A265155   b(1) =  3 -> {3, 6, 25, 281, 961, ...},    which quickly merges into A090566
%e A265155     (as does the result of using b(1) = 6 or 12 or 20 ...);
%e A265155   b(1) =  5 -> {5, 29, 241, 1809, ...},      which quickly merges into A265147
%e A265155     (as does the result of using b(1) = 7 ...);
%e A265155   b(1) =  9 -> {9, 61, 504, 4516, ...},      which quickly merges into A265148;
%e A265155   b(1) = 13 -> {13, 69, 169, 744, 769, ...}, which quickly merges into A265151.
%t A265155 (* See the Mmca coding in A090566 or A265147-A265154. *)
%Y A265155 Cf. A090566, A243091, A265147, A265148, A265149, A265150, A265151, A265152, A265153, A265154.
%K A265155 nonn,base,more
%O A265155 1,2
%A A265155 _Robert G. Wilson v_, Dec 02 2015