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A071520 Number of 5-smooth numbers (A051037) <= n.

%I A071520 #39 Apr 29 2025 04:44:28
%S A071520 1,2,3,4,5,6,6,7,8,9,9,10,10,10,11,12,12,13,13,14,14,14,14,15,16,16,
%T A071520 17,17,17,18,18,19,19,19,19,20,20,20,20,21,21,21,21,21,22,22,22,23,23,
%U A071520 24,24,24,24,25,25,25,25,25,25,26,26,26,26,27,27,27,27,27,27,27,27,28,28
%N A071520 Number of 5-smooth numbers (A051037) <= n.
%C A071520 A 5-smooth number is a number of the form 2^x*3^y*5^z (x,y,z) >= 0.
%H A071520 Amiram Eldar, <a href="/A071520/b071520.txt">Table of n, a(n) for n = 1..10000</a>
%F A071520 a(n) = Card{ k | A051037(k) <= n }.
%F A071520 Asymptotically : let a = 1/(6*log(2)*log(3)*log(5)) and b = sqrt(30) then a(n) = a*log(b*n)^3 + O(log(n)).
%F A071520 a(n) = -Sum_{k=1,n} mu(30*k)*floor(n/k). - _Benoit Cloitre_, Jun 14 2007
%F A071520 a(n) = Sum_{i=0..floor(log_5(n))} Sum_{j=0..floor(log_3(n/5^i))} floor(log_2(2*n/(5^i*3^j))). - _Ridouane Oudra_, Jul 17 2020
%t A071520 Accumulate[Table[If[Max[FactorInteger[n][[;;,1]]]<6,1,0],{n,80}]] (* _Harvey P. Dale_, Aug 04 2024 *)
%o A071520 (PARI) for(n=1,100,print1(sum(k=1,n,if(sum(i=4,n,if(k%prime(i),0,1)),0,1)),","))
%o A071520 (PARI) a(n)=-sum(k=1,n,moebius(2*3*5*k)*floor(n/k)) \\ _Benoit Cloitre_, Jun 14 2007
%o A071520 (Python)
%o A071520 from sympy import integer_log
%o A071520 def A071520(n):
%o A071520     c = 0
%o A071520     for i in range(integer_log(n,5)[0]+1):
%o A071520         for j in range(integer_log(m:=n//5**i,3)[0]+1):
%o A071520             c += (m//3**j).bit_length()
%o A071520     return c # _Chai Wah Wu_, Sep 16 2024
%Y A071520 Cf. A008683, A051037, A106598, A112751.
%Y A071520 Number of p-smooth numbers <= n: A070939 (p=2), A071521 (p=3), this sequence (p=5), A071604 (p=7), A071523 (p=11), A080684 (p=13), A080685 (p=17), A080686 (p=19).
%K A071520 easy,nonn
%O A071520 1,2
%A A071520 _Benoit Cloitre_, Jun 02 2002
%E A071520 Title corrected by _Rainer Rosenthal_, Aug 30 2020