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A385137 The sum of divisors d of n such that n/d is a 3-smooth number (A003586).

%I A385137 #8 Jun 20 2025 08:09:21
%S A385137 1,3,4,7,5,12,7,15,13,15,11,28,13,21,20,31,17,39,19,35,28,33,23,60,25,
%T A385137 39,40,49,29,60,31,63,44,51,35,91,37,57,52,75,41,84,43,77,65,69,47,
%U A385137 124,49,75,68,91,53,120,55,105,76,87,59,140,61,93,91,127,65,132
%N A385137 The sum of divisors d of n such that n/d is a 3-smooth number (A003586).
%H A385137 Amiram Eldar, <a href="/A385137/b385137.txt">Table of n, a(n) for n = 1..10000</a>
%F A385137 a(n) = A064987(n)/A385138(n).
%F A385137 Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if p <= 3, and p^e if p >= 5.
%F A385137 In general, the sum of divisors d of n such that n/d is q-smooth (not divisible by a prime larger than q) is multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if p <= q, and p^e if p > q.
%F A385137 Dirichlet g.f.: zeta(s-1) / ((1 - 1/2^s) * (1 - 1/3^s)).
%F A385137 In general, the sum of divisors d of n such that n/d is q-smooth has Dirichlet g.f.: zeta(s-1) / Product_{p prime <= q} (1 - 1/q^s).
%F A385137 Sum_{k=1..n} a(k) ~ (3/4)*n^2.
%F A385137 In general, the sum of divisors d of n such that n/d is prime(k)-smooth has an average order c * n^2 / 2, where c = A072044(k-1)/A072045(k-1) for k >= 2.
%t A385137 f[p_, e_] := If[p < 5, (p^(e+1) - 1)/(p - 1), p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A385137 (PARI) a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(p < 5, (p^(e + 1) - 1)/(p - 1), p^e));}
%Y A385137 Cf. A003586, A064987, A072044, A072045, A072079.
%Y A385137 The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), A385134 (biquadratefree), A385135 (exponentially odd), A385136 (cubefull), this sequence (3-smooth), A385138 (5-rough), A385139 (exponentially 2^n).
%K A385137 nonn,easy,mult
%O A385137 1,2
%A A385137 _Amiram Eldar_, Jun 19 2025