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a(n) is the dimension of the multilinear part of the free flexible Lie-admissible algebra with n generators.

Cf. A370677 for flexible Lie-admissible algebras.

allocated for Paul Laubiea(n) is the dimension of the multilinear part of the free flexible Lie-admissible algebra with n generators.

1, 2, 9, 61, 545, 5986

1,2

The flexible identity is (x*y)*z + (z*y)*x = x*(y*z) + z*(y*x).

Flexible algebras are algebras (possibly non-associative) satisfying this identity.

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nonn,hard,more

Paul Laubie, Feb 26 2024

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allocated for Paul Laubie

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Numerator of fraction a(n)/b(n) < 3, where a(n) and b(n) are primes, and if p is a prime <= b(n) such that k/p < 3 for some prime k, then k = a(n) and p = b(n).

5, 13, 19, 31, 37, 67, 109, 127, 139, 157, 181, 199, 211, 307, 337, 379, 409, 487, 499, 541, 571, 577, 631, 751, 769, 787, 811, 829, 877, 919, 937, 991, 1009, 1039, 1117, 1201, 1291, 1297, 1327, 1381, 1399, 1459, 1471, 1567, 1621, 1669, 1759, 1777, 1801

1,1

(a(n)/(b(n)) is a strictly increasing sequence that converges to 3.

Apparently, a(n) = A091180(n) for n > 1. - Hugo Pfoertner, Feb 20 2024

5/2 < 13/5 < 19/7 < 31/11 < 37/13 < 67/23 < 109/37 < 127/43 < 139/47 < ...

a = {{4, 4}}; r = 3; Do[If[# < Last[a][[2]], AppendTo[a, {n, #}]] &[

NextPrime[Prime[PrimePi[Prime[n]*r]]]/Prime[n]], {n, 1300}];

Numerator[Rest[Map[#[[2]] &, a]]] (* this sequence *)

Denominator[Rest[Map[#[[2]] &, a]]] (* A370162 *)

(* Peter J. C. Moses, Feb 06 2024 *)

Cf. A000040, A088878, A091180, A370162.

nonn,frac,new

recycled

Clark Kimberling, Feb 16 2024

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Apparently, a(n) = A091180(n) for n > 1. - Hugo Pfoertner, Feb 20 2024

Cf. A000040, A088878, A091180, A370162.

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editing