A346465 Numbers k such that (4^k - 2)*(4^k - 1)/Clausen(2*k, 1) is not squarefree, where Clausen(n, m) = A160014(n, m).

9, 11, 18, 27, 32, 36, 45, 50, 53, 54, 63, 68, 72, 74, 78, 81, 90, 95, 99, 100, 108, 116, 117, 126, 127, 135, 137, 144, 147, 150, 153, 155, 158, 162, 171, 179, 180, 182, 189, 198, 200, 204, 207, 216, 221, 225, 233, 234, 242, 243, 250, 252, 261, 263, 270, 279

Offset: 1

Author

Peter Luschny, Jul 20 2021

Keywords

• nonn

Comments

Also numbers k such that 6*GaussBinomial(2*k, 2, 2)/denominator(Bernoulli(2*k, 1)) is not squarefree.

Crossrefs

Cf. A006095, A002445, A007947, A160014, A346463, A346464.

Programs

Maple

with(NumberTheory): isa := n -> not IsSquareFree(((4^n - 2)*(4^n - 1))/
mul(i, i = select(isprime, map(i -> i+1, Divisors(2*n))))):
select(isa, [$(1..100)]);

Mathematica

q[n_] := Product[k, {k, Select[Table[d + 1, {d, Divisors[2 n]}], PrimeQ]}];
isA[n_] := ! SquareFreeQ[((4^n - 2) (4^n -1)) / q[n]];
Select[Range[50],  isA]

Formula

The positive multiples of 9 form a subsequence.

k is a term if and only if A346463(k) > A007947(A346463(k)).

Extensions

More terms from Jinyuan Wang, Jul 23 2021