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%I #17 Oct 05 2021 20:24:48
%S 9,11,18,27,32,36,45,50,53,54,63,68,72,74,78,81,90,95,99,100,108,116,
%T 117,126,127,135,137,144,147,150,153,155,158,162,171,179,180,182,189,
%U 198,200,204,207,216,221,225,233,234,242,243,250,252,261,263,270,279
%N Numbers k such that (4^k - 2)*(4^k - 1)/Clausen(2*k, 1) is not squarefree, where Clausen(n, m) = A160014(n, m).
%C Also numbers k such that 6*GaussBinomial(2*k, 2, 2)/denominator(Bernoulli(2*k, 1)) is not squarefree.
%F The positive multiples of 9 form a subsequence.
%F k is a term if and only if A346463(k) > A007947(A346463(k)).
%p with(NumberTheory): isa := n -> not IsSquareFree(((4^n - 2)*(4^n - 1))/
%p mul(i, i = select(isprime, map(i -> i+1, Divisors(2*n))))):
%p select(isa, [$(1..100)]);
%t q[n_] := Product[k, {k, Select[Table[d + 1, {d, Divisors[2 n]}], PrimeQ]}];
%t isA[n_] := ! SquareFreeQ[((4^n - 2) (4^n -1)) / q[n]];
%t Select[Range[50], isA]
%Y Cf. A006095, A002445, A007947, A160014, A346463, A346464.
%K nonn
%O 1,1
%A _Peter Luschny_, Jul 20 2021
%E More terms from _Jinyuan Wang_, Jul 23 2021
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