A317848 Multiplicative with a(p^e) = binomial(2*e, e).

1, 2, 2, 6, 2, 4, 2, 20, 6, 4, 2, 12, 2, 4, 4, 70, 2, 12, 2, 12, 4, 4, 2, 40, 6, 4, 20, 12, 2, 8, 2, 252, 4, 4, 4, 36, 2, 4, 4, 40, 2, 8, 2, 12, 12, 4, 2, 140, 6, 12, 4, 12, 2, 40, 4, 40, 4, 4, 2, 24, 2, 4, 12, 924, 4, 8, 2, 12, 4, 8, 2, 120, 2, 4, 12, 12, 4, 8, 2, 140

Offset: 1

Author

Andrew Howroyd, Aug 08 2018

Keywords

• nonn
• mult

Comments

The Dirichlet convolution square of this sequence is A165825.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A000265, A000984, A006519, A037445, A046643, A046644, A165825, A299149, A299150.

Programs

Mathematica

f[p_, e_] := Binomial[2*e, e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 30 2023 *)

PARI

a(n)={my(v=factor(n)[,2]); prod(i=1, #v, binomial(2*v[i], v[i]))}

PARI

\\ DirSqrt(v) finds u such that v = v[1]*dirmul(u, u).
DirSqrt(v)={my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&d
				

PARI

A317848(n) = factorback(apply(e -> binomial(e+e,e),factor(n)[,2])); \\ Antti Karttunen, Sep 17 2018

Formula

A037445(n) = A006519(a(n)).

A046643(n) = numerator(a(n)/A165825(n)) = A000265(a(n)).

A046644(n) = denominator(a(n)/A165825(n)) = A165825(n)/A037445(n).

A299149(n) = numerator(n*a(n)/A165825(n)) = A000265(n*a(n)).

A299150(n) = denominator(n*a(n)/A165825(n)) = A165825(n)/(A037445(n) * A006519(n)).