A368885 - OEIS

%I #13 Jan 10 2024 00:29:42

%S 1,1,1,2,1,1,1,1,2,1,1,2,1,1,1,1,1,2,1,2,1,1,1,1,2,1,1,2,1,1,1,1,1,1,

%T 1,4,1,1,1,1,1,1,1,2,2,1,1,1,2,2,1,2,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,2,

%U 1,1,1,2,1,1,2,2,1,1,1,1,1,1,1,2,1,1,1

%N The number of unitary divisors of n that are squares of a squarefree number (A062503).

%C First differs from A294932 at n = 32.

%C The largest of these divisors is A368884(n).

%H Amiram Eldar, <a href="/A368885/b368885.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = 2 if e = 2, and 1 otherwise.

%F a(n) >= 1, with equality if and only if n is in A337050.

%F a(n) <= A034444(n), with equality if and only if n is in A062503.

%F Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(2*s) - 1/p^(3*s)).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/p^2 - 1/p^3) = 1.30596827416754083231... .

%t f[p_, e_] := If[e == 2, 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = vecprod(apply(x->if(x==2, 2, 1), factor(n)[, 2]));

%o (Python)

%o from sympy import factorint

%o def A368885(n): return 1<<sum(1 for e in factorint(n).values() if e==2) # _Chai Wah Wu_, Jan 09 2024

%Y Cf. A034444, A062503, A294932, A337050, A368884.

%K nonn,easy,mult

%O 1,4

%A _Amiram Eldar_, Jan 09 2024