A365335 - OEIS

%I #14 Sep 11 2023 07:38:19

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

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

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

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

%C First differs from A160338 at n = 64, and from A178489 at n = 65.

%C The number of divisors of the square root of the largest square dividing n is A046951(n).

%C The number of exponentially odd divisors of the square root of the largest square dividing n is A365549(n) and their sum is A365336(n). [corrected, Sep 08 2023]

%H Amiram Eldar, <a href="/A365335/b365335.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A325837(A000188(n)).

%F a(n) > 1 if and only if n is a bicubeful number (A355265).

%F Multiplicative with a(p^e) = floor((e+2)/4).

%F Dirichlet g.f.: zeta(s) * zeta(4*s) * Product_{p prime} (1 - 1/p^(4*s) + 1/p^(6*s)).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(4) * Product_{p prime} (1 - 1/p^4 + 1/p^6) = 1.0181534831085... .

%t f[p_, e_] := Max[1, Floor[(e+2)/4]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = vecprod(apply(x -> max(1, (x+2)\4), factor(n)[, 2]));

%Y Cf. A000188, A046951, A325837, A355265, A365336, A365549.

%Y Cf. A160338, A178489.

%K nonn,easy,mult

%O 1,64

%A _Amiram Eldar_, Sep 01 2023

%E Name corrected by _Amiram Eldar_, Sep 08 2023