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A385046 The sum of the unitary divisors of n that are 3-smooth numbers (A003586).

%I A385046 #8 Jun 16 2025 16:53:14
%S A385046 1,3,4,5,1,12,1,9,10,3,1,20,1,3,4,17,1,30,1,5,4,3,1,36,1,3,28,5,1,12,
%T A385046 1,33,4,3,1,50,1,3,4,9,1,12,1,5,10,3,1,68,1,3,4,5,1,84,1,9,4,3,1,20,1,
%U A385046 3,10,65,1,12,1,5,4,3,1,90,1,3,4,5,1,12,1,17
%N A385046 The sum of the unitary divisors of n that are 3-smooth numbers (A003586).
%C A385046 The number of these divisors is A382488(n), and the largest of them is A065331(n).
%H A385046 Amiram Eldar, <a href="/A385046/b385046.txt">Table of n, a(n) for n = 1..10000</a>
%F A385046 Multiplicative with a(p^e) = p^e + 1 if p <= 3, and 1 if p >= 5.
%F A385046 a(n) = A034448(n)/A385045(n).
%F A385046 a(n) <= A034448(n), with equality if and only if n 3-smooth.
%F A385046 a(n) <= A072079(n).
%F A385046 Dirichlet g.f.: zeta(s) * ((1-1/2^(2*s-1))/(1-1/2^(s-1))) * ((1-1/3^(2*s-1))/(1-1/3^(s-1))).
%F A385046 Sum_{k=1..n} a(k) ~ (n/(6*log(2)*log(3))) * (log(n)^2 + c1*log(n) + c2), where c1 = 2*gamma - 2 + 7*log(2) + 5*log(3) - 2*log(6) = 5.916004..., c2 = 2 - 5*log(2) - 11*log(2)^2/6 - 3*log(3) - 5*log(3)^2/6 + 15*log(2)*log(3)/2 + (5*log(2) + 3*log(3) - 2)*gamma - 2*gamma_1 = 1.957142..., gamma is Euler's constant (A001620), and gamma_1 is the 1st Stieltjes constant (A082633).
%t A385046 f[n_, p_] := If[Divisible[n, p], p^IntegerExponent[n, p] + 1, 1]; a[n_] := f[n, 2]*f[n, 3]; Array[a, 100]
%o A385046 (PARI) a(n) = if(n%2, 1, 2^valuation(n, 2)+1) * if(!(n%3), 3^valuation(n, 3)+1, 1);
%Y A385046 The unitary analog of A072079.
%Y A385046 Cf. A001620, A003586, A034448, A065331, A082633, A382488.
%Y A385046 The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), A385043 (exponentially 2^n), A385045 (5-rough), this sequence (3-smooth), A385047 (power of 2), A385048 (cubefull), A385049 (biquadratefree).
%K A385046 nonn,easy,mult
%O A385046 1,2
%A A385046 _Amiram Eldar_, Jun 16 2025