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Integers which are unique starting points for the algorithm described in A090566.
8

%I #16 Aug 30 2021 21:54:16

%S 1,2,4,8,10,11,14,15,16,17,18,19,21,22,23

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

%C 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 b(1) = a(1) = 1 yields {1, 6, 25, 281, 961, ...} (A090566),

%C b(1) = a(2) = 2 yields {2, 5, 29, 241, 1809, ...} (A265147),

%C b(1) = a(3) = 4 yields {4, 9, 61, 504, 4516, ...} (A265148),

%C b(1) = a(4) = 8 yields {8, 41, 209, 764, 5225, ...} (A265149),

%C b(1) = a(5) = 10 yields {10, 24, 336, 400, 689, ...} (A265150),

%C b(1) = a(6) = 11 yields {11, 56, 169, 744, 769, ...} (A265151),

%C ...

%e 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 b(1) = 3 -> {3, 6, 25, 281, 961, ...}, which quickly merges into A090566

%e (as does the result of using b(1) = 6 or 12 or 20 ...);

%e b(1) = 5 -> {5, 29, 241, 1809, ...}, which quickly merges into A265147

%e (as does the result of using b(1) = 7 ...);

%e b(1) = 9 -> {9, 61, 504, 4516, ...}, which quickly merges into A265148;

%e b(1) = 13 -> {13, 69, 169, 744, 769, ...}, which quickly merges into A265151.

%t See the Mmca coding in A090566 or A265147-A265154.

%Y Cf. A090566, A243091, A265147, A265148, A265149, A265150, A265151, A265152, A265153, A265154.

%K nonn,base,more

%O 1,2

%A _Robert G. Wilson v_, Dec 02 2015