cp's OEIS Frontend

A368169 The number of divisors of the largest unitary divisor of n that is a cubefull exponentially odd number (A368167).

%I A368169 #7 Dec 16 2023 09:01:43
%S A368169 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 A368169 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 A368169 1,1,1,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A368169 The number of divisors of the largest unitary divisor of n that is a cubefull exponentially odd number (A368167).
%H A368169 Amiram Eldar, <a href="/A368169/b368169.txt">Table of n, a(n) for n = 1..10000</a>
%F A368169 Multiplicative with a(p^e) = e+1 if e is odd that is larger than 1, and 1 otherwise.
%F A368169 a(n) = A000005(A368167(n)).
%F A368169 a(n) >= 1, with equality if and only if n is in A335275.
%F A368169 a(n) <= A000005(n), with equality if and only if n is in A335988.
%F A368169 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 A368169 f[p_, e_] := If[e == 1 || EvenQ[e], 1, e+1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A368169 (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 A368169 Cf. A000005, A013661, A098198, A335275, A335988, A368167, A368168.
%K A368169 nonn,easy,mult
%O A368169 1,8
%A A368169 _Amiram Eldar_, Dec 14 2023