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A355582 a(n) is the largest 5-smooth divisor of n.

%I A355582 #24 Apr 20 2025 03:30:55
%S A355582 1,2,3,4,5,6,1,8,9,10,1,12,1,2,15,16,1,18,1,20,3,2,1,24,25,2,27,4,1,
%T A355582 30,1,32,3,2,5,36,1,2,3,40,1,6,1,4,45,2,1,48,1,50,3,4,1,54,5,8,3,2,1,
%U A355582 60,1,2,9,64,5,6,1,4,3,10,1,72,1,2,75,4,1,6,1,80
%N A355582 a(n) is the largest 5-smooth divisor of n.
%H A355582 Amiram Eldar, <a href="/A355582/b355582.txt">Table of n, a(n) for n = 1..10000</a>
%H A355582 Vaclav Kotesovec, <a href="/A355582/a355582.jpg">Graph - the asymptotic ratio (1000000 terms)</a>
%F A355582 Multiplicative with a(p^e) = p^e if p <= 5 and 1 otherwise.
%F A355582 a(n) = A006519(n) * A038500(n) * A060904(n).
%F A355582 a(n) = 2^A007814(n) * 3^A007949(n) * 5^A112765(n).
%F A355582 a(n) = n / A165725(n).
%F A355582 Dirichlet g.f.: zeta(s)*(2^s-1)*(3^s-1)*(5^s-1)/((2^s-2)*(3^s-3)*(5^s-5)). - _Amiram Eldar_, Dec 25 2022
%F A355582 Sum_{k=1..n} a(k) ~ 2*n*log(n)^3 / (45*log(2)*log(3)*log(5)) + O(n*log(n)^2). - _Vaclav Kotesovec_, Apr 20 2025
%t A355582 a[n_] := Times @@ ({2, 3, 5}^IntegerExponent[n, {2, 3, 5}]); Array[a, 100]
%o A355582 (PARI) a(n) = 3^valuation(n, 3) * 5^valuation(n, 5) << valuation(n, 2);
%o A355582 (Python)
%o A355582 from sympy import multiplicity as v
%o A355582 def a(n): return 2**v(2, n) * 3**v(3, n) * 5**v(5, n)
%o A355582 print([a(n) for n in range(1, 81)]) # _Michael S. Branicky_, Jul 08 2022
%Y A355582 Cf. A006519, A007814, A007949, A038500, A051037, A060904, A065331, A112765, A132741, A165725, A355583, A355584.
%Y A355582 Cf. A379005 (rgs-transform), A379006 (ordinal transform).
%K A355582 nonn,mult,easy
%O A355582 1,2
%A A355582 _Amiram Eldar_, Jul 08 2022