A081063 - OEIS

%I #14 Sep 16 2024 12:48:09

%S 1,2,3,4,5,6,7,8,9,9,10,11,12,12,12,13,14,15,16,16,16,16,17,18,19,19,

%T 20,20,21,21,22,23,23,23,23,24,25,25,25,25,26,26,27,27,27,27,28,29,30,

%U 30,30,30,31,32,32,32,32,32,33,33,34,34,34,35,35,35,36,36,36,36,37,38,39

%N Number of numbers <= n that are 3-smooth or prime powers.

%C a(n) = #{A081061(k): 1<=k<=n}.

%H Harvey P. Dale, <a href="/A081063/b081063.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n)=A071521(n)+A065515(n)-A000523(n)-A062153(n)+1.

%t Accumulate[Table[If[PrimePowerQ[n]||Max[FactorInteger[n][[All,1]]]<5,1,0],{n,80}]] (* _Harvey P. Dale_, Nov 29 2020 *)

%o (Python)

%o from sympy import integer_log, primepi, integer_nthroot

%o def A081063(n): return int(1-(a:=n.bit_length())-(b:=integer_log(n,3)[0])+sum((n//3**i).bit_length() for i in range(b+1))+sum(primepi(integer_nthroot(n, k)[0]) for k in range(1, a))) # _Chai Wah Wu_, Sep 16 2024

%Y Cf. A003586, A000961.

%K nonn

%O 1,2

%A _Reinhard Zumkeller_, Mar 04 2003