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A385138 The sum of divisors d of n such that n/d is a 5-rough number (A007310).

%I A385138 #8 Jun 20 2025 08:09:43
%S A385138 1,2,3,4,6,6,8,8,9,12,12,12,14,16,18,16,18,18,20,24,24,24,24,24,31,28,
%T A385138 27,32,30,36,32,32,36,36,48,36,38,40,42,48,42,48,44,48,54,48,48,48,57,
%U A385138 62,54,56,54,54,72,64,60,60,60,72,62,64,72,64,84,72,68,72
%N A385138 The sum of divisors d of n such that n/d is a 5-rough number (A007310).
%H A385138 Amiram Eldar, <a href="/A385138/b385138.txt">Table of n, a(n) for n = 1..10000</a>
%F A385138 a(n) = A064987(n)/A385137(n).
%F A385138 Multiplicative with a(p^e) = p^e if p <= 3, and (p^(e+1)-1)/(p-1) if p >= 5.
%F A385138 In general, the sum of divisors d of n such that n/d is q-rough (not divisible by a prime smaller than q) is multiplicative with a(p^e) = p^e if p <= q, and (p^(e+1)-1)/(p-1) if p > q.
%F A385138 Dirichlet g.f.: zeta(s-1) * zeta(s) * ((1 - 1/2^s) * (1 - 1/3^s)).
%F A385138 In general, the sum of divisors d of n such that n/d is q-rough has Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime <= q} (1 - 1/q^s).
%F A385138 Sum_{k=1..n} a(k) ~ (Pi^2/18)*n^2.
%F A385138 In general, the sum of divisors d of n such that n/d is prime(k)-rough has an average order c * n^2 / 2, where c = zeta(2) * A072045(k-1)/A072044(k-1) for k >= 2.
%t A385138 f[p_, e_] := If[p > 3, (p^(e+1) - 1)/(p - 1), p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A385138 (PARI) a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(p > 3, (p^(e + 1) - 1)/(p - 1), p^e));}
%Y A385138 Cf. A007310, A013661, A064987, A072044, A072045, A186099.
%Y A385138 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), A385137 (3-smooth), this sequence (5-rough), A385139 (exponentially 2^n).
%K A385138 nonn,easy,mult
%O A385138 1,2
%A A385138 _Amiram Eldar_, Jun 19 2025