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

%I A317848 #15 Apr 30 2023 02:08:33
%S A317848 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,
%T A317848 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,
%U A317848 2,24,2,4,12,924,4,8,2,12,4,8,2,120,2,4,12,12,4,8,2,140
%N A317848 Multiplicative with a(p^e) = binomial(2*e, e).
%C A317848 The Dirichlet convolution square of this sequence is A165825.
%H A317848 Antti Karttunen, <a href="/A317848/b317848.txt">Table of n, a(n) for n = 1..65537</a>
%F A317848 A037445(n) = A006519(a(n)).
%F A317848 A046643(n) = numerator(a(n)/A165825(n)) = A000265(a(n)).
%F A317848 A046644(n) = denominator(a(n)/A165825(n)) = A165825(n)/A037445(n).
%F A317848 A299149(n) = numerator(n*a(n)/A165825(n)) = A000265(n*a(n)).
%F A317848 A299150(n) = denominator(n*a(n)/A165825(n)) = A165825(n)/(A037445(n) * A006519(n)).
%t A317848 f[p_, e_] := Binomial[2*e, e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Apr 30 2023 *)
%o A317848 (PARI) a(n)={my(v=factor(n)[,2]); prod(i=1, #v, binomial(2*v[i], v[i]))}
%o A317848 (PARI) \\ DirSqrt(v) finds u such that v = v[1]*dirmul(u, u).
%o A317848 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<n, u[d]*u[n/d], 0)))/2); u}
%o A317848 DirSqrt(vector(80, n, 4^bigomega(n)))
%o A317848 (PARI) A317848(n) = factorback(apply(e -> binomial(e+e,e),factor(n)[,2])); \\ _Antti Karttunen_, Sep 17 2018
%Y A317848 Cf. A000265, A000984, A006519, A037445, A046643, A046644, A165825, A299149, A299150.
%K A317848 nonn,mult
%O A317848 1,2
%A A317848 _Andrew Howroyd_, Aug 08 2018