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The number of divisors of the largest 4th power dividing n.
1

%I #12 Aug 08 2024 14:28:08

%S 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,5,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,5,1,1,

%T 1,1,1,1,1,1,1,1,1,1,1,1,1,5,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,5,1,1,1,1,

%U 1,1,1,1,1,1,1,1,1,1,1,5,5,1,1,1,1,1,1

%N The number of divisors of the largest 4th power dividing n.

%C The number of divisors of the 4th root of the largest 4th power dividing n, A053164(n), is A063775(n).

%H Amiram Eldar, <a href="/A365490/b365490.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A000005(A008835(n)).

%F Multiplicative with a(p^e) = 4*floor(e/4) + 1.

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

%F a(n) <= A000005(n) with equality if and only if n is a fourth power (A000583).

%F Dirichlet g.f.: zeta(s) * zeta(4*s) * Product_{p prime} (1 + 3/p^(4*s)).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(4) * Product_{p prime} (1 + 3/p^4) = 1.3414590511076... . In general, the asymptotic mean of the number of divisors of the largest k-th power dividing n is zeta(k) * Product_{p prime} (1 + (k-1)/p^k).

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

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

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def A365490(n): return prod(e&-4|1 for e in factorint(n).values()) # _Chai Wah Wu_, Aug 08 2024

%Y Cf. A000005, A000583, A046100, A008835, A053164, A063775.

%K nonn,easy,mult

%O 1,16

%A _Amiram Eldar_, Sep 05 2023