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A385044 The number of unitary divisors of n that are 5-rough numbers (A007310).

%I A385044 #8 Jun 16 2025 16:53:21
%S A385044 1,1,1,1,2,1,2,1,1,2,2,1,2,2,2,1,2,1,2,2,2,2,2,1,2,2,1,2,2,2,2,1,2,2,
%T A385044 4,1,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,1,4,2,2,2,2,2,2,2,2,1,4,2,2,2,
%U A385044 2,4,2,1,2,2,2,2,4,2,2,2,1,2,2,2,4,2,2
%N A385044 The number of unitary divisors of n that are 5-rough numbers (A007310).
%C A385044 The sum of these divisors is A385045(n), and the largest of them is A065330(n).
%H A385044 Amiram Eldar, <a href="/A385044/b385044.txt">Table of n, a(n) for n = 1..10000</a>
%F A385044 Multiplicative with a(p^e) = 1 if p <= 3, and 2 if p >= 5.
%F A385044 a(n) = A034444(n)/A382488(n).
%F A385044 a(n) <= A034444(n), with equality if and only if n is 5-rough.
%F A385044 a(n) <= A035218(n).
%F A385044 Dirichlet g.f.: (zeta(s)^2/zeta(2*s)) * (1/((1+1/2^s)*(1+1/3^s))).
%F A385044 Sum_{k=1..n} a(k) ~ (n / (2 * zeta(2))) *(log(n) + 2*gamma - 1 + log(2)/3 + log(3)/4 - 2*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620).
%t A385044 f[p_, e_] := If[p <= 3, 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A385044 (PARI) a(n) = vecprod(apply(x -> if(x <= 3, 1, 2), factor(n)[, 1]));
%Y A385044 The unitary analog of A035218.
%Y A385044 Cf. A007310, A034444, A065330, A385045.
%Y A385044 Cf. A001620, A013661, A306016.
%Y A385044 The number of unitary divisors of n that are: A000034 (power of 2), A055076 (exponentially odd), A056624 (square), A056671 (squarefree), A068068 (odd), A323308 (powerful), A365498 (cubefree), A365499 (biquadratefree), A368248 (cubefull), A380395 (cube), A382488 (3-smooth), A385042 (exponentially 2^n), this sequence (5-rough).
%K A385044 nonn,easy,mult
%O A385044 1,5
%A A385044 _Amiram Eldar_, Jun 16 2025