A365333 - OEIS

%I #15 Sep 11 2023 07:38:37

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

%T 1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,

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

%N The number of exponentially odd coreful divisors of the largest square dividing n.

%C First differs from A043289, A053164, A063775, A203640 and A295658 at n = 64.

%C The number of squares dividing the largest exponentially odd divisor of n is A325837(n).

%C The sum of the exponentially odd divisors of the largest square dividing n is A365334(n). [corrected, Sep 08 2023]

%C The number of exponentially odd divisors of the largest square dividing n is the same as the number of squares dividing n, A046951(n). - _Amiram Eldar_, Sep 08 2023

%H Amiram Eldar, <a href="/A365333/b365333.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A325837(A008833(n)).

%F a(n) = 1 if and only if n is a biquadratefree number (A046100).

%F Multiplicative with a(p^e) = max(1, floor(e/2)).

%F Dirichlet g.f.: zeta(s) * zeta(4*s) * zeta(6*s) / zeta(12*s).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 15015/(1382*Pi^2) = 1.100823... .

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

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

%Y Cf. A008833, A046100, A046951, A325837, A365334.

%Y Cf. A043289, A053164, A063775, A203640, A295658.

%K nonn,easy,mult

%O 1,16

%A _Amiram Eldar_, Sep 01 2023

%E Name corrected by _Amiram Eldar_, Sep 08 2023