A071520 - OEIS

%I #32 Sep 17 2024 04:31:00

%S 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 17,17,17,18,18,19,19,19,19,20,20,20,20,21,21,21,21,21,22,22,22,23,23,

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

%C A 5-smooth number is a number of the form 2^x*3^y*5^z (x,y,z) >= 0.

%F a(n) = Card{ k | A051037(k) <= n }.

%F 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 a(n) = -Sum_{k=1,n} mu(30*k)*floor(n/k). - _Benoit Cloitre_, Jun 14 2007

%F 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 Accumulate[Table[If[Max[FactorInteger[n][[;;,1]]]<6,1,0],{n,80}]] (* _Harvey P. Dale_, Aug 04 2024 *)

%o (PARI) for(n=1,100,print1(sum(k=1,n,if(sum(i=4,n,if(k%prime(i),0,1)),0,1)),","))

%o (PARI) a(n)=-sum(k=1,n,moebius(2*3*5*k)*floor(n/k)) \\ _Benoit Cloitre_, Jun 14 2007

%o (Python)

%o from sympy import integer_log

%o def A071520(n):

%o     c = 0

%o     for i in range(integer_log(n,5)[0]+1):

%o         for j in range(integer_log(m:=n//5**i,3)[0]+1):

%o             c += (m//3**j).bit_length()

%o     return c # _Chai Wah Wu_, Sep 16 2024

%Y Cf. A051037, A106598, A112751.

%K easy,nonn

%O 1,2

%A _Benoit Cloitre_, Jun 02 2002

%E Title corrected by _Rainer Rosenthal_, Aug 30 2020