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a(n) is the least number that starts a sequence of exactly n nonsquarefree numbers under the iteration x_{n+1} = x_n^2 - x_n + 1.
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%I #17 Feb 06 2022 23:03:52

%S 4,68,1832,80796

%N a(n) is the least number that starts a sequence of exactly n nonsquarefree numbers under the iteration x_{n+1} = x_n^2 - x_n + 1.

%C 1000000 < a(5) <= 58229352.

%e a(3) = 1832 because 1832 is divisible by 2^2, 1832^2-1832+1 = 3354393 is divisible by 7^2, 3354393^2-3354393+1 = 11251949044057 is divisible by 97^2, and 11251949044057^2-11251949044057+1 = 126606357290043984177975193 is squarefree, and 1832 is the smallest number that works.

%p f:= proc(n) local x,i;

%p x:= n;

%p for i from 0 do

%p if numtheory:-issqrfree(x) then return count fi;

%p x:= x^2-x+1

%p od

%p end proc:

%p W:= Vector(4):

%p for n from 1 while W[4] = 0 do

%p v:= f(n);

%p if v >= 1 and W[v] = 0 then W[v]:= n fi

%p od:

%p convert(W,list);

%Y Cf. A013929.

%K nonn,more

%O 1,1

%A _Robert Israel_, Feb 03 2022