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A385045 The sum of the unitary divisors of n that are 5-rough numbers (A007310).

%I A385045 #7 Jun 16 2025 16:54:02
%S A385045 1,1,1,1,6,1,8,1,1,6,12,1,14,8,6,1,18,1,20,6,8,12,24,1,26,14,1,8,30,6,
%T A385045 32,1,12,18,48,1,38,20,14,6,42,8,44,12,6,24,48,1,50,26,18,14,54,1,72,
%U A385045 8,20,30,60,6,62,32,8,1,84,12,68,18,24,48,72,1,74,38
%N A385045 The sum of the unitary divisors of n that are 5-rough numbers (A007310).
%C A385045 First differs from A186099 at n = 25; a(25) = 26, while A186099(25) = 31.
%C A385045 The number of these divisors is A385044(n), and the largest of them is A065330(n).
%H A385045 Amiram Eldar, <a href="/A385045/b385045.txt">Table of n, a(n) for n = 1..10000</a>
%F A385045 Multiplicative with a(p^e) = 1 if p <= 3, and p^e + 1 if p >= 5.
%F A385045 a(n) = A034448(n)/A385046(n).
%F A385045 a(n) <= A034448(n), with equality if and only if n is 5-rough.
%F A385045 a(n) <= A186099(n).
%F A385045 Dirichlet g.f.: (zeta(s)*zeta(s-1)/zeta(2*s-1)) * ((1-1/2^(s-1))/(1-1/2^(2*s-1))) * ((1-1/3^(s-1))/(1-1/3^(2*s-1))).
%F A385045 Sum_{k=1..n} a(k) ~ c * n^2, where c = 3*Pi^2/(91*zeta(3)) = 0.270679... .
%t A385045 f[p_, e_] := If[p <= 3, 1, p^e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A385045 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] < 5, 1, f[i, 1]^f[i, 2] + 1));}
%Y A385045 The unitary analog of A186099.
%Y A385045 Cf. A002117, A007310, A034448, A065330, A385044.
%Y A385045 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), this sequence (5-rough), A385046 (3-smooth), A385047 (power of 2), A385048 (cubefull), A385049 (biquadratefree).
%K A385045 nonn,easy,mult
%O A385045 1,5
%A A385045 _Amiram Eldar_, Jun 16 2025