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The number of divisors of the largest unitary divisor of n that is a cubefull exponentially odd number (A368167).
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%I #7 Dec 16 2023 09:01:43

%S 1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,1,1,4,1,1,1,1,6,1,1,

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

%U 1,1,1,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1

%N The number of divisors of the largest unitary divisor of n that is a cubefull exponentially odd number (A368167).

%H Amiram Eldar, <a href="/A368169/b368169.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = e+1 if e is odd that is larger than 1, and 1 otherwise.

%F a(n) = A000005(A368167(n)).

%F a(n) >= 1, with equality if and only if n is in A335275.

%F a(n) <= A000005(n), with equality if and only if n is in A335988.

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 3/p^3 - 2/p^4 - 1/p^5 + 1/p^6) = 1.47140789970892803631... .

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

%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1 || !(f[i, 2]%2), 1, f[i, 2]+1));}

%Y Cf. A000005, A013661, A098198, A335275, A335988, A368167, A368168.

%K nonn,easy,mult

%O 1,8

%A _Amiram Eldar_, Dec 14 2023